Chapter 18: Homeworks
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Homework I
Definition for time scales. Jump operators and graininess functions
Check if the following sets are time scales. Provide your answers with detailed explanations.
1. \(\{-1, 2, 3,9, 12\}\).
2. \(C\cup [0, 1]\), where \(C\) is the Cantor set.
3. \((-7, -3]\cup [-2, 0]\cup [7, 15]\).
4. \(\left\{\frac{1}{10^n}\right\}_{n\in \mathbb{N}_0}\cup \{0\}\cup 7^{\mathbb{N}_0}\).
5. \(3^{\mathbb{N}_0}\cup [-3, 0]\).
6. \(4\mathbb{Z}\).
7. \(5^{\mathbb{N}_0}\).
8. \((1-U)\cup (1+U)\cup (2-U)\cup (2+U)\), where \(U=\left\{\frac{1}{n}\right\}_{n\in \mathbb{N}}\).
9. \(\{-10, -8, -5, 0, 7, 14\}\cup [15, \infty)\).
10. \(\left(-3^{\mathbb{N}_0}\right)\cup 5^{\mathbb{N}_0}\).
11. \(\left(-7^{\mathbb{N}_0}\right)\cup \mathbb{N}_0\).
12. \([-10, 0]\cup \{2, 34, 56, 78\}\).
13. \(\left\{\frac{1}{n^4}\right\}_{n\in \mathbb{N}}\cup \{0\}\).
14. \([-7, -5]\cup 3^{\mathbb{N}_0}\).
15. \((-5, -4]\cup [-3, 0]\cup (1, 12]\).
16. \((-10, -9)\cup \{-8, -7, -6, -5, -4, -3, -2, -1, 0\}\).
17. \(11^{\mathbb{N}_0}\).
18. \([-3, 0]\cup \{2\}\).
19. \([-10, -8]\cup [0, 2]\).
20. \([-3, 0]\cup 5^{\mathbb{N}_0}\).
Classify each of the points of the following time scales and find their forwardjump operators, backward jump operators, forward graineness functions and backward graininess functions.
1. \((2h+1)\mathbb{Z}\), where \(h>0\).
2. \(4^{\mathbb{N}_0}\).
3. \(\mathbb{N}_0^4\).
4. \(\{3H_n: n\in \mathbb{N}_0\}\), where \(H_n\), \(n\in \mathbb{N}_0\), are the harmonic numbers.
5. \([0, 1]\cup [4, 5]\cup [8, 9]\cup [12, 13]\cup\ldots\).
6. \(\left\{\sum\limits_{j=0}^n \frac{1}{j^2+1}: n\in \mathbb{N}_0\right\}\).
7. \(\left\{ \left(\frac{1}{3}\right)^{3^n}\right\}_{n\in \mathbb{N}}\cup\{0, 2\}\).
8. \(3\mathbb{Z}+2\).
9. \(-4\mathbb{N}_0+7\).
10. \((-2\mathbb{N}_0)\cup 3^{\mathbb{N}_0}\).
11. \(P_{6, 10}\cup [3, 4]\).
12. \(\{0\}\cup 9^{\mathbb{N}_0}\).
13. \([1, 2]\cup [3, 4]\cup [7, 8]\cup 9^{\mathbb{N}}\).
14. \((-5\mathbb{N}_0)\cup 3^{\mathbb{N}_0}\).
15. \([-1, 3]\cup [7, 9]\cup [10, \infty)\).
16. \(\{0\}\cup \left\{ 1-\frac{1}{4^n}\right\}_{n\in \mathbb{N}_0}\cup 2^{\mathbb{N}_0}\).
17. \(\{2\}\cup \left\{2+\frac{1}{n}\right\}_{n\in \mathbb{N}}\cup [4, 9]\).
18. \([1, 3]\cup 5^{\mathbb{N}}\).
19. \([-1, 0]\cup [2, 3]\cup 4^{\mathbb{N}}\).
20. \(3^{\mathbb{N}_0}\cup 9^{\mathbb{N}_0}\).
Find \(\sigma(t)\), \(t\in \mathbb{T}\), where \(\mu(t)\), \(t\in \mathbb{N}\), is a solution of the following equations
1. \(\frac{\mu(t)}{\mu(t)-5}=\frac{\mu(t)-5}{\mu(t)-6}\), \(\mu(t)\ne 5, 6\).
2. \((\mu(t))^2-15\mu(t)+26=0\).
3. \((\mu(t))^2-9\mu(t)+90=0\).
4. \(\frac{\mu(t)-1}{(\mu(t))^2-25}+\frac{2\mu(t)-8}{\mu(t)+5}=\frac{\mu(t)-1}{2\mu(t)+10}-\frac{1}{5-\mu(t)}\).
5. \(\frac{3\mu(t)-1}{\mu(t)+7}-\frac{3(\mu(t))^2-7}{49-(\mu(t))^2}=\frac{8}{\mu(t)+7}-\frac{\mu(t)-1}{\mu(t)-7}\).
6. \(\frac{3}{(\mu(t))^2+\mu(t)-2}=\frac{1}{\mu(t)(\mu(t)-1)^2}+\frac{3}{\mu(t)(\mu(t)-3)}\).
7. \((\mu(t)+1)(\mu(t)-4)(\mu(t)+5)(\mu(t)-8)=225\).
8. \((\mu(t))^2+2|\mu(t)-1|-6=0\).
9. \( |(\mu(t))^2+3|+|\mu(t)+2|=8\).
10.\( \frac{\mu(t)+1}{|\mu(t)-1|}-5\frac{|\mu(t)-1|}{\mu(t)+1}+4=0\).
11. \( \frac{4\mu(t)}{4(\mu(t))^2-8\mu(t)=7}+\frac{3\mu(t)}{4(\mu(t))^2-10\mu(t)+7}=1\).
12. \(\sqrt{(\mu(t))^2-7}=3\).
13. \(3+\sqrt{\mu(t)-6}=5\).
14. \( \sqrt{(\mu(t))^2-3\mu(t)+4}=\sqrt{2}\).
15. \(3^{\mu(t)+2}+3^{\mu(t)}=270\).
16. \( 5^{\mu(t)-1}-5^{\mu(t)-3}=120\).
17. \(7^{2\mu(t)}+7^{2\mu(t)-2}-7^{2\mu(t)-3}=385\).
18. \(3\cdot 2^{\mu(t)+1}+5\cdot 2^{\mu(t)+2}-2^{\mu(t)+3}=36\).
19. \(\log_{2\mu(t)-1}(4(\mu(t))^2-5\mu(t)+5)=2\).
20. \( \log_{3\mu(t)+2}(3(\mu(t))^2+17\mu(t)+10)=2\).
Find \(\rho(t)\), \(t\in \mathbb{T}\), where \(\nu(t)\), \(t\in \mathbb{T}\) is a solution of the following equations.
1. \(\frac{\nu(t)+2}{\nu(t)-5}=\frac{\nu(t)+1}{\nu(t)-2}\).
2. \(\frac{2\nu(t)+3}{4\nu(t)-1}=\frac{3\nu(t)-5}{46+6\nu(t)}\).
3. \(\frac{\nu(t)-1}{3\nu(t)+3}-\frac{\nu(t)-2}{3\nu(t)+5}=0\).
4. \(\frac{3\nu(t)}{3\nu(t)+2}-\frac{\nu(t)-2}{\nu(t)-1}=0\).
5. \(\frac{3\nu(t)-1}{\nu(t)+7}-\frac{3(\nu(t))^2-7}{49-(\nu(t))^2}=\frac{8}{\nu(t)+7}+\frac{\nu(t)-1}{\nu(t)+7}\).
6. \(\frac{1-(\nu(t))^2}{4(\nu(t))^2-1}+\frac{5\nu(t)-4}{2\nu(t)+1}=\frac{P8-0\nu(t)}{1-2\nu(t)}\).
7. \(\frac{\nu(t)}{\nu(t)-3}+\frac{9}{(\nu(t))^2-9\nu(t)+18}=\frac{2}{\nu(t)-6}\).
8. \(\frac{1}{(\nu(t))^2-5\nu(t)+6}+\frac{5-2\nu(t)}{\nu(t)-2}=1+\frac{\nu(t)+6}+\frac{5-2\nu(t)}{\nu(t)-3}\).
9. \(\frac{\nu(t)+5}{3\nu(t)+2}-\frac{\nu(t)+6}{\nu(t)-2}=\frac{(\nu(t))^2+\nu(t)+5}{3(\nu(t))^2-4\nu(t)-4}\).
10. \(\frac{1}{\nu(t)+6}+\frac{1}{\nu(t)+7}-\frac{1}{\nu(t)+9}-\frac{1}{\nu(t)+10}=\frac{21}{20}\).
11. ((\nu(t))^2-6\nu(t))^2-2(\nu(t)-3)^2=81\).
12. \(\frac{(\nu(t))^2+2\nu(t)+1}{(\nu(t))^2+2\nu(t)+2}+\frac{(\nu(t))^2+2\nu(t)+2}{(\nu(t))^2+2\nu(t)+3}=\frac{7}{6}\).
13. \(\nu(t)+\sqrt{65-(\nu(t))^2}=9\).
14. \( 2\nu(t)-\sqrt{(\nu(t))^2-137}=5\).
15. \(\sqrt{3\nu(t)+3}-3\nu(t)=1\).
16. \( \sqrt{3\nu(t)+10}-\nu(t)=4\).
17. \(\sqrt{3(\nu(t))^2-4\nu(t)+9}-3=\nu(t)\).
18. \(\log\sqrt{(\nu(t))^2+2\nu(t)+1}=\log 7\).
19. \(\log((\nu(t))^2-8\nu(t)+4)=2\log(\nu(t)-6)\).
20. \(4\cdot 7^{\nu(t)-2}=28^{\nu(t)+1}\).
Find \(\sigma(t)\) and \(\rho(t)\) for \(t\in \mathbb{T}\), where \(\mu(t), \nu(t)\), \(t\in \mathbb{T}\), are solutions of the follwoing systems.
1.
\[\begin{eqnarray*}\frac{5}{\mu(t)} -\frac{9}{\nu(t)}&=& 3\frac{1}{6}\\ \frac{1}{\mu(t)}+\frac{6}{\nu(t)}&=& 2\frac{1}{3}.\end{eqnarray*}\notag\]
2.
\[\begin{eqnarray*} \frac{4}{\mu(t)-2}-\frac{1}{\nu(t)-3}&=& 1\\ \frac{7}{\mu(t)-2}+\frac{5}{\nu(t)-3}&=& 8\frac{1}{2}.\end{eqnarray*}\notag\]
3.
\[\begin{eqnarray*} \frac{1}{\mu(t)+\nu(t)}+\frac{1}{\mu(t)-\nu(t)}&=& \frac{1}{3}\\ \frac{1}{\mu(t)+\nu(t)}-\frac{1}{\mu(t)-\nu(t)}&=& \frac{1}{6}.\end{eqnarray*}\notag\]
4.
\[\begin{eqnarray*}\frac{2}{\mu(t)+\nu(t)}+\frac{1}{\mu(t)-\nu(t)}&=& \frac{5}{4}\\ \frac{3}{\mu(t)+\nu(t)}-\frac{4}{\mu(t)-\nu(t)}&=& \frac{1}{2}.\end{eqnarray*}\notag\]
5.
\[\begin{eqnarray*} \frac{3}{4\mu(t)+3\nu(t)}+\frac{2}{4\mu(t)-3\nu(t)}&=& \frac{37}{55}\\ \frac{5}{4\mu(t)+3\nu(t)}-\frac{1}{4\mu(t)-3\nu(t)}&=& \frac{14}{55}.\end{eqnarray*}\notag\]
6.
\[\begin{eqnarray*} \frac{6}{2\mu(t)+\nu(t)-1}-\frac{2}{2\mu(t)-\nu(t)+3}&=& \frac{5}{2}\\ \frac{4}{2\mu(t)+\nu(t)-1}-\frac{4}{\nu(t)-2\mu(t)-3}&=& 3.\end{eqnarray*}\notag\]
7.
\[\begin{eqnarray*} \frac{\nu(t)+1}{\mu(t)-\nu(t)}+\frac{\mu(t)+2}{\mu(t)+\nu(t)}&=& \frac{(\mu(t))^2+(\nu(t))^2+10}{(\mu(t))^2-(\nu(t))^2}\\ 2\mu(t)+5\nu(t)&=& 1.\end{eqnarray*}\notag\]
8.
\[\begin{eqnarray*} \frac{5}{(\mu(t))^2+5\mu(t)\nu(t)}+\frac{7}{\mu(t)\nu(t)+5(\nu(t))^2}-\frac{2}{\mu(t)\nu(t)}&=& \frac{10}{(\mu(t))^2\nu(t)+5\mu(t)(\nu(t))^2}\\ \frac{3\mu(t)-\nu(t)-10}{2}&=& \frac{3\mu(t)+\nu(t)-15}{3}.\end{eqnarray*}\notag\]
9.
\[\begin{eqnarray*} \frac{\nu(t)-2}{\mu(t)+2\nu(t)}-\frac{\mu(t)-3}{\mu(t)+2\nu(t)}-\frac{\mu(t)-3}{\mu(t)-2\nu(t)}&=& \frac{(\mu(t))^2+\mu(t)\nu(t)+2(\nu(t))^2-1}{4(\nu(t))^2-(\mu(t))^2}\\ 3\mu(t)-11\nu(t)&=& 85.\end{eqnarray*}\notag\]
10.
\[\begin{eqnarray*}\frac{\mu(t)-2}{\mu(t)+2}+\frac{\nu(t)+4}{\nu(t)-2}&=& \frac{\mu(t)(1-\mu(t)-\nu(t))}{4-(\mu(t))^2}\\ \frac{3-\mu(t)+\nu(t)}{6}&=& \frac{\nu(t)-\mu(t)}{5}.\end{eqnarray*}\notag\]
11.
\[\begin{eqnarray*} \frac{2}{5\mu(t)-\nu(t)-3}&=& \frac{\mu(t)-\nu(t)+60}{25(\mu(t))^2-9-6\nu(t)-(\nu(t))^2}\\ 2\mu(t)+\nu(t)&=& 13.\end{eqnarray*}\notag\]
12.
\[\begin{eqnarray*} \frac{\mu(t)+\nu(t)}{\mu(t)\nu(t)-\mu(t)-]\nu(t)+1}&=& \frac{2\mu(t)+\nu(t)+5}{(\mu(t))^2+\mu(t)\nu(t)-\mu(t)-\nu(t)}-\frac{\mu(t)+7}{\mu(t)+\nu(t)-\mu(t)\nu(t)-(\nu(t))^2}\\ \frac{2\mu(t)-18\nu(t)+23}{9(\mu(t))^2-16+6\mu(t)\nu(t)+(\nu(t))^2}&=& \frac{3}{3\mu(t)+\nu(t)+4}.\end{eqnarray*}\notag\]
13.
\[\begin{eqnarray*} \frac{1}{\mu(t)}+\frac{1}{\nu(t)}&=& 3\\ 2\mu(t)+3\nu(t)&=& 7\mu(t)\nu(t).\end{eqnarray*}\notag\]
14.
\[\begin{eqnarray*} |2\mu(t)-1|-\nu(t)&=& 2\\ \mu(t)-|4-\nu(t)|&=& -1.\end{eqnarray*}\notag\]
15.
\[\begin{eqnarray*} |\mu(t)|+3\nu(t)&=& 7\\ 2\mu(t)+2|\nu(t)-1|&=& 3.\end{eqnarray*}\notag\]
16.
\[\begin{eqnarray*} |\mu(t)-3|+2|\nu(t)-1|&=& 2\\ \mu(t)+|\nu(t)-1|&=& \frac{9}{2}.\end{eqnarray*}\notag\]
17.
\[\begin{eqnarray*} (\mu(t))^2+(\nu(t))^2+\mu(t)-\nu(t)&=& 4\\ (\mu(t))^2-(\nu(t))^2+2\mu(t)+\nu(t)&=& 1.\end{eqnarray*}\notag\]
18.
\[\begin{eqnarray*}(\mu(t))^2+(\nu(t))^2&=& 5\\ (\mu(t))^2-(\nu(t))^2&=& -1\end{eqnarray*}\notag\]
19.
\[\begin{eqnarray*} 2(\mu(t))^2+(\nu(t))^2&=& 17\\ (\nu(t))^2-(\mu(t))^2&=& 5.\end{eqnarray*}\notag\]
20.
\[\begin{eqnarray*} 5(\mu(t))^2-3(\nu(t))^2&=& 2\\ 3(\mu(t))^2-2(\nu(t))^2&=& 1.\end{eqnarray*}\]
Let \(\mathbb{T}=\left\{\frac{1}{2^n}\right\}_{n\in n\mathbb{N}_0}\cup\{0\}\cup 3^{\mathbb{N}_0}\). Find
\[f^\sigma(t),\quad f^\rho(t),\quad f^{\sigma\rho}(t),\quad f^{\rho\sigma}(t),\quad t\in \mathbb{T},\]
where
1. \(f(t)=3t^2-6t+5\), \(t\in \mathbb{T}\).
2. \(f(t)=5t^2-10t+3\), \(t\in \mathbb{T}\).
3. \(f(t)=-3t^2+60t-17\), \(t\in \mathbb{T}\).
4. \(f(t)=\frac{1}{3}t^3-2t^2-5t+6\), \(t\in \mathbb{T}\).
5. \(f(t)=20+24t-3t^2-t^3\), \(t\in \mathbb{T}\).
6. \(f(t)=\frac{2t+5}{5-2t}\), \(t\in \mathbb{T}\).
7.\( f(t)=\frac{t}{3}+\frac{3}{t}\), \(t\in \mathbb{T}\).
8. \(f(t)=2t^3+6t^2-18t+120\), \(t\in \mathbb{T}\).
9. \( f(t)=t^3-2t^2+32t+18\), \(t\in \mathbb{T}\).
10. \(f(t)=\frac{t+3}{t^2-10}\), \(t\in \mathbb{T}\).
11. \( f(t)=t^4-8t^2-9\), \(t\in \mathbb{T}\).
12. \(f(t)=t^3-3t^2+3\), \(t\in \mathbb{T}\).
13. \(f(t)=\frac{1}{3}t^3-2t^2+3t+1\), \(t\in \mathbb{T}\).
14. \(f(t)= \frac{t-1}{t+3}\), \(t\in \mathbb{T}\).
15. \(f(t)= -\frac{2t+1}{3t+4}\), \(t\in \mathbb{T}\).
16. \(f(t)=\frac{t}{t^2+4}\), \(t\in \mathbb{T}\).
17. \(f(t)=t^2-8t+7\), \(t\in \mathbb{T}\).
18. \(f(t)= t^3-3t^2-24t+50\), \(t\in \mathbb{T}\).
19. \(f(t)= t^4-4t^2+5\), \(t\in \mathbb{T}\).
20. \(f(t)=t^2(t-3)\), \(t\in \mathbb{T}\).
Simplify
1. \(\frac{\sigma(t)+\frac{1}{\sigma(t)}}{\sigma(t)+3}\).
2. \(\frac{2\sigma(t)-15}{(\sigma(t))^2}\).
3. \(\left(\frac{1}{2\sigma(t)+8}-\frac{3-\sigma(t)}{3\sigma(t)-6}\right): \frac{12-\sigma(t)}{\sigma(t)}\).
4. \(\left(\frac{5+\sigma(t)}{\sigma(t)-3}+\frac{\sigma(t)}{2\sigma(t)-1}\right)\cdot \frac{\sigma(t)+1}{\sigma(t)}\).
5. \(\left(\frac{4\sigma(t)-1}{(\sigma(t))^2+4}+\frac{2-\frac{3}{\sigma(t)-1}}{\sigma(t)+2}\right)\cdot \left(1+\frac{3}{\sigma(t)}\right)\).
6. \(\left(\frac{5-\sigma(t)}{2(\sigma(t))^2+3}-\frac{\sigma(t)}{2}+1\right): \frac{(\sigma(t))^2+10}{3}\).
7. \( \frac{3}{|\sigma(t)-1|}+\frac{2\sigma(t)}{(\sigma(t))^2+1}\).
8. \(\left(\frac{3-\sigma(t)}{|\sigma(t)|+4}-\frac{\sigma(t)}{2(\sigma(t))^4+5}\right): \frac{(\sigma(t))^2+1}{3}\).
9. \(\frac{1}{(\sigma(t))^2-\sigma(t)}+\frac{2}{1-(\sigma(t))^2}+\frac{1}{(\sigma(t))^2+\sigma(t)}\).
10.\( \frac{1}{6\sigma(t)+3}-\frac{\frac{16}{3}\sigma(t)+3}{8(\sigma(t))^2-2}+\frac{1}{2\sigma(t)-1}\).
11. \(\frac{(\nu(t))^2-\nu(t)-6}{(\nu(t))^2-4}-\frac{\nu(t)-1}{2-\nu(t)}-2\).
12. \(\frac{2}{\nu(t)+4}+\frac{\nu(t)-9}{16-(\nu(t))^2}-\frac{\nu(t)-3}{(\nu(t))^2-8\nu(t)+16}\).
13. \(\frac{\nu(t)}{2\nu(t)-1}+2\frac{\nu(t)-1}{2\nu(t)}-\frac{1}{2\nu(t)-4(\nu(t))^2}-\frac{2\nu(t)-1}{\nu(t)+1}\).
14. \(\left(1-\frac{3(\nu(t))^2}{1-(\nu(t))^2}\right): \left(\frac{\nu(t)}{\nu(t)-1}+1\right)\).
15. \( \frac{1}{1+\frac{x}{1-\frac{x}{x+2}}}: \frac{\frac{1}{1-x}+\frac{1}{1+x}}{\frac{1}{1-x}-\frac{1}{1+x}}\).
16. \(\left( \frac{2\sigma(t)\nu(t)-1}{(\sigma(t))^2-16}+\frac{\nu(t)}{\sigma(t)+4}\right): \frac{\sigma(t)+6}{\sigma(t)-6}-\frac{\nu(t)}{(\sigma(t))^2+9}\).
17.\( \left(\frac{\sigma(t)-2\nu(t)}{\sigma(t)+2\nu(t)}+\frac{2-\nu(t)}{\sigma(t)}\right)\cdot \frac{1}{2\nu(t)}-(\sigma(t)+2\nu(t)):\frac{\nu(t)+1}{\nu(t)-1}\).
18. \(\left(\frac{\sigma(t)}{\sigma(t)-3\nu(t)}+\frac{\nu(t)}{\sigma(t)+3\nu(t)}\right)\cdot \frac{(\sigma(t))^2+6\sigma(t)\nu(t)+9(\nu(t))^2}{(\sigma(t))^2+4\sigma(t)\nu(t)-3(\nu(t))^2}\).
19. \(\left( \frac{6\sigma(t)+\nu(t)}{(\sigma(t))^2-6\sigma(t)\nu(t)}+\frac{6\sigma(t)-\nu(t)}{(\sigma(t))^2+6\sigma(t)\nu(t)}\right)\cdot \frac{(\sigma(t)0^2-36(\nu(t))^2}{(\sigma(t))^2+(\nu(t))^2}\).
20. \(\left(\frac{\sigma(t)+4\nu(t)}{2\nu(t)}-\frac{6\nu(t)}{4\nu(t)-\sigma(t)}\right)\cdot \left(1-\frac{(\sigma(t))^2-2\sigma(t)\nu(t)+4(\nu(t))^2}{(\sigma(t)0^2-4(\nu(t))^2}\right)\).
Homework II
Delta and Nabla Differentiation
Let \(\mathbb{T}=\left\{\frac{1}{n}\right\}_{n\in \mathbb{N}}\cup\{0\}\cup 2^{\mathbb{N}}\). Find \(f^\Delta(t)\), \(t\in \mathbb{T}^\kappa\), and \(f^{\Delta^2}(t)\), \(t\in \mathbb{T}^{\kappa^2}\), where
1. \(f(t)=t^2-3t+2\).
2.\( f(t)=\frac{2}{3}t-t^3\).
3. \(f(t)=\frac{t-1}{t}\).
4. \(f(t)=\frac{t^2-3t+2}{t^2+t+1}\).
5. \(f(t)=t^4-t^3+\frac{1}{(t+2)^2}\).
6. \(f(t)=(t^2+10(t-3)\).
7. \(f(t)=\frac{t}{t^2+1}+t^3-t\).
8. \(f(t)=(t-1)(t^2+3)\).
9. \(f(t)= \frac{t}{2}+\frac{2}{t}\).
10. \(f(t)=(t+2)^2-5t^2\).
11. \( f(t)=t(t^2-3)(t^2+1)\).
12. \(f(t)=(2t^2-t)\sqrt{t}\).
13. \(f(t)=\sqrt[3]{t}-\frac{1}{2}t^2+\sqrt{t}\).
14. \(f(t)=\frac{\sqrt{t}}{t+3}\).
15. \(f(t)= \frac{5+t}{t-3}\).
16. \(f(t)= t+\frac{t+2}{t+4}\).
17. \(f(t)= \frac{t^2-3t]{t^2+4}\).
18. \(f(t)= \frac{t^2-t+1}{t^2+t+3}\).
19. \(f(t)= (t^2-3t+\sqrt{2})^3\).
20. \(f(t)=\left(\frac{2-t}{3+t}\right)^2\).
21. \(f(t)= \sqrt{Pt^2-3}\).
22. \(f(t)=\sqrt{7t-t^2-12}\).
23. \(f(t)=t+2\sqrt{t}\).
24. \(f(t)= \frac{t^5}{t^3-2}\).
25. \(f(t)=\frac{t^3-3t^2+1}{t-1}\).
26. \(f(t)= (t^4-1)+\frac{1}{t+1}\).
27. \(f(t)=(t+1)(t+2)(t+3)(t+4)\).
28. \(f(t)=\frac{1}{t^4+1}-t^2\).
29. \(f(t)=t^2-(t+1)(t+2)^2\).
30. \(f(t)=\frac{t+1}{t+2}-t^4-\sqrt{t}\).
Let \(\mathbb{T}=\left\{1-\frac{1}{3^n}\right\}_{n\in \mathbb{N}_0}\cup 4^{\mathbb{N}_0}\). Find \(f^\nabla(t)\), \(t\in \mathbb{T}_\kappa\), and \(f^{\nabla^2}(t)\), \(t\in \mathbb{T}_{\kappa^2}\), where
1. \(f(t)= 5t^2-3t+1\).
2. \(f(t)=12-t^2\).
3. \(f(t)=\frac{3}{2-t}\).
4. \(f(t)= t^3-3t-2\).
5. \(f(t)= 4t^3-21t^2+18t+20\).
6. \(f(t)= \frac{1}{2}t^4-\frac{4}{3}t^3-24t^2+7\).
7. \(f(t)= 20+24t-3t^2-t^3\).
8. \(f(t)=\frac{2t+5}{5-2t}\).
9. \(f(t)=\frac{t}{3}+\frac{3}{t}\).
10. \(f(t)=\frac{t+3}{t^2+10}\).
11. \(f(t)= \frac{t^2-2t+2}{t+1}\).
12. \(f(t)=\frac{16}{t(4-t^2)}\).
13. \(f(t)=\frac{4}{\sqrt{t^2+1}}\).
14. \(f(t)=\frac{3}{t-2}-\frac{5}{t+2}\).
15. \(f(t)= \frac{t^2-24}{4-t^2}\).
16. \(f(t)= \frac{1}{t^2+2t+2}\).
17. \(f(t)= \frac{t^2+t+1}{t^2-t+1}\).
18. \(f(t)= \frac{t-1}{\sqrt[5]{t}+1}\).
19. \(f(t)=\frac{1}{t(t+1)}\).
20. \(f(t)= \frac{2t+3}{t^2-5t+6}\).
21. \(f(t)=(t+1)(t+2)^2\).
22. \(f(t)=\frac{1}{t}+\frac{2}{t^2}|+\frac{3}{t^3}\).
23. \(f(t)=4t^3-3t^5\).
24. \(f(t)=\frac{t+3}{2t+9}-t^2\).
25. \(f(t)=(t+3)(t+5)(t+7)\).
26. \(f(t)=\frac{1}{t^2+7t+30}-t(t+2)(t+3)\).
27. \(f(t)=t^4-(t+2)(t+5`)\).
28. \(f(t)=t^7\).
29. \(f(t)=t^4+4t^3+8t^2+6t+5\).
30. \(f(t)=t^3-\frac{3}{(t+2)^2}\).
Let \(\mathbb{T}=\left\{1-\frac{1}{4^n}\right\}_{n\in \mathbb{N}_0}\cup \{1\}\cup 5^{\mathbb{N}}\). Find the intrervals where the following functions increase and decrease.
1. \(f(t)=13t+1\).
2. \(f(t)=5-3t\).
3. \(f(t)= 5t^2-3t+1\).
4. \(f(t)=12-t^2\).
5. \(f(t)=(t-1)^2\).
6. \(f(t)=\frac{2}{t}\).
7. \( f(t)=\frac{3}{2-t}\).
8. \(f(t)= t^3-3t-2\).
9. \(f(t)=4t^3-21t^2+18t+20\).
10. \( f(t)= \frac{1}{2}t^4-\frac{4}{3}t^3-24t^2+7\).
11. \(f(t)=\frac{2t-1}{t^2+t+3}\).
12. \(f(t)=12t+5\).
13. \(f(t)=5t^2-10t+3\).
14. \(f(t)=5t^2+10t-3\).
15. \(f(t)=-3t^2+60t+27\).
16. \(f(t)= 2\sqrt{t^2+t-20}\).
17. \(f(t)=\frac{1}{3}t^3-2t^2-5t+6\).
18. \(f(t)-=20+24t-3t^2-t^3\).
19. \(f(t)=\frac{2t+5}{5-2t}\).
20. \(f(t)= \frac{t+3}{t^2-20}\).
21. \(f(t)= \frac{t-1}{t+3}\).
22. \(f(t)= -\'frac{2t+1}{3t+4}\).
23. \(f(t)= \frac{t}{t^2+4}\).
24. \(f(t)= t^3-3t^2-24t+50\).
25. \(f(t)=t^4-4t^2+5\).
26. \(f(t)= t^2(t-3)\).
27. \(f(t)=t^2(t-3)\).
28. \(f(t)= \frac{t^2-3t+11}{t+8}\).
29. \(f(t)=\frac{4}{\sqrt{t^2+1}}\).
30. \(f(t)= \frac{1}{(t+3)(t^2-25)}\).
Let \(\mathbb{T}=[-3, 1]\cup \{2\}\cup 3^{\mathbb{N}}\). Find the local extreme points of the following functions.
1. \(f(t)= gt^3-12t\).
2. \(f(t)= t^3-9t^2+15t-2\).
3. \(f(t)= 1+(t-2)^3\).
4. \(f(t)= t^5-5t^4+5t^2-1\).
5. \(f(t)= t^3-3t^2+6t+7\).
6. \(f(t)=1+(t-1)^4\).
7. \(f(t)= t^4-8t^3+22t^2-24t+12\).
8. \(f(t)= (t-4)^4(t-3)^4\).
9. \(f(t)= \frac{t}{1+t^2}\).
10. \(f(t)= t+\frac{1}{t}\).
11. \(f(t)= \frac{1}{t}+\frac{1}{2-t^2}\).
12. \(f(t)= \frac{t^2-7t+6}{t-10}\).
13. \(f(t)= \frac{t^4+1}{t^2}\).
14. \(f(t)= \frac{t^2}{t^4+4}\).
15. \(f(t)=t^3(t-1)^{2\over 3}\).
16. \(f(t)= \frac{1+t^2}{1+t^4}\).
17. \(f(t)=3t^2-t^3\).
18. \(f(t)=\frac{1}{1+t^2}\).
19. \(f(t)= t^4-8t^3-2\).
20. \(f(t)= 2t^3-=9t^2+t-3\).
21. \(f(t)= (t+1)(t+2)^3\).
22. \(f(t)= \frac{1}{t+1}+\frac{2}{(t+1)^2}+\frac{3}{(t+1)^3}\).
23. \(f(t)= \frac{t}{\sqrt{3-t^2}}\).
24. \(f(t)= 2t^3+3t^2+2t-7\).
25. \(f(t)= -t(t-2)^3\).
Let \(\mathbb{T}=[-2, 0]\cup\{1\}\cup 3^{\mathbb{N}_0}\). Investigate for convexity and concavity the following functions.
1. \(f(t)=\frac{2}{3}t^3-t^2-4t-5\).
2. \(f(t)= 3t^4-8t^3+6t^2+1\).
3. \(f(t)= 2t^3-6t^2-18t+7\).
4. \(f(t)= t^4+4t^3-8t^2+3\).
5. \(f(t)=4t^4-2t^2+3\).
6. \(f(t)=\frac{t^2}{t^2+3}\).
7. \(f(t)=\frac{(t-2)(8-t)}{t^2}\).
8. \(f(t)= (t-1)(t-2)(t-3)\).
9. \(f(t)= 4t^4-3t^3\).
10. \(f(t)=t^4-10t^2+9\).
11. \(f(t)= (t-3)^2(t-2)^3\).
12. \(f(t)=\frac{1}{5}t^5-4t^2\).
13. \(f(t)= t^5-t^2+8\).
14. \(f(t)= \frac{t^2-3t+2}{t^2+3t+2}\).
15. \(f(t)=-t^3+3t^2+5\).
16. \(f(t)= 3t^3-9t^2+2\).
17. \(f(t)=t^3-4t^2+4t-3\).
18. \(f(t)= t^2(2t-3)-12(3t-2)\).
19. \(f(t)= \frac{1}{15}t^3+\frac{9}{20}t^2-t+1\).
20. \(f(t)=|t^2-4t+3|+2t\).
21. \(f(t)=\frac{1}{2+t}{3+2t}-t\).
22. \(f(t)=\frac{1}{t^2+1}-t\).
23. \(f(t)=t^5-4t^3+t^2+1\).
24. \(f(t)= 2(t+1)^3-t^2\).
25. \(f(t)=t^5-1\).
Homework III
Delta and Nabla Integration. Elementary Functions
Find \(z_1\oplus_h z_2\), where
1. \(z_1=-1, z_2=3+i, h=2\).
2. \(z_1=1+2i, z_2=1+i, h=4\).
3. \(z_1=-2, z_2=i, h=3\).
4. \(z_1=2-4i, z_2=3+i, h=2\).
5. \(z_1=1+5i, z_2=2-i, h=3\).
6. \(z_1=2, z_2=4, h=3\).
7. \(z_1=2-i, z_2=1+i, h=4\).
8. \(z_1=2+i, z_2=(1+i)^3, h=3\).
9. \(z_1=3-2i, z_2=(2-i)^4, h=5\).
10. \(z_1=2+i(i-2), z_2=(3-i)(1+2i)-2, h=7\).
11. \(z_1=2-i, z_2=4+i, h=5\).
12. \(z_1=(1-3i)^2, z_2=-1+i, h=3\).
13. \(z_1=i, z_2=(1-2i)^2, h=3\).
14. \(z_1=1+2i, z_2=(3-i)^3, h=4\).
15. \(z_1=\frac{i-1}{i-2}, z_2=(1+i)(1-3i)-2, h=5\).
Find \(z_1\ominus_h z_2\), where
1. \(z_1=2+i, z_2=3-i, h=4\).
2. \(z_1=(1+i)(7-i), z_2=3-i, h=3\).
3. \(z_1=(4+i)^2, z_2=(1-i)(2-8i), h=4\).
4. \(z_1=2, z_2, 3, h=-1\).
5. \(z_1=(1+i)^2, z_2=(2-i)^3, h=3\).
6. \(z_1=(2-3i)^3, z_2=1+3i, h=3\).
7. \(z_1=i, z_2=\frac{2+i}{7-i}, h=4\).
8. \(z_1=\frac{1-i}{2-5i}, z_2=2+i, h=4\).
9. \(z_1=2-71i, z_2=4, h=4\).
10. \(z_1=4i, z_2=3i, h=9\).
11. \(z_1=\frac{2+i}{7-i}, z_2=1+i, h=3\).
12. \(z_1=\frac{i}{i+2}, z_2=\frac{1+i}{3-2i}, h=5\).
13. \(z_1=\frac{1+i}{2+5i}, z_2=2+7i, h=3\).
14. \(z_1=i, z_2=(4i)^3, h=5\).
15. \(z_1=(i-1)^3, z_2=i, h=8\).
Let \(\mathbb{T}=[-2, 0]\cup 2^{\mathbb{N}_0}\). Find
\[ (f\oplus_\mu g)(t)\ominus_\nu (f(t)-2g(t)),\quad t\in \mathbb{T},\notag\]
where
1. \(f(t)=t, g(t)=2t\).
2. \(f(t()=t^2+1, g(t)=t\).
3. \(f(t)=3-t, g(t)=2t+3\).
4. \(f(t)=\frac{1+t}{1+2t}, g(t)=t^2+1\).
5. \(f(t)=1+t+t^2, g(t)=t-1\).
6. \(f(t)=1-3t, g(t)=1+t^2\).
7. \(f(t)=\frac{1+3t}{1+5t}, g(t)=t-1\).
8. \(f(t)=t, g(t)=1+t-2t^2\).
9. \(f(t)=1-t+t^2, g(t)=t-3\).
10. \(f(t)=2t+1, g(t)=3t-7\).
Let \(\mathbb{T}=(-\mathbb{N}_0)\cup [1, 2]\cup 3^{\mathbb{N}}\). Determine if the following functions are rd-continuous, ld-continuous and regulated.
1. \(f(t)=\frac{1}{3}t^3-2\).
2. \(f(t)=t^2-3t+4\).
3. \(f(t)=\frac{1+t}{2+3t}\).
4. \(f(t)=t^3-3t\).
5. \(f(t)= \frac{1}{t^2+1}-t\).
6. \(f(t)=t^3-t^2+\frac{1}{t^2+1}\).
7. \(f(t)=t^2-\frac{1}{t^4+1}\).
8. \(f(t)=\frac{1-t}{1+t}-3t^2-t+3\).
9. \(f(t)=1+t-(1+2t)(3+5t)\).
10. \(f(t)=t^2-\frac{1}{t-1}\).
Let \(\mathbb{T}=[0, 2]\cup 3^{\mathbb{N}}\). Compute the following integrals
1. \(\int (2t+1)\Delta t\).
2. \(\int (3t^2+2t-1)\Delta t\).
3. \(\int\left(\sqrt{t}+\frac{1}{2t}\right)\Delta t\).
4. \(\int t^2(t^2+1)\Delta t\).
5. \(\int\frac{t^2-3t+4}{\sqrt{t}}\Delta t\).
6. \(\int \frac{1}{t^2(1-t^2)}\Delta t\).
7. \(\int\frac{t^2}{1-t^2}\Delta t\).
8. \(\int \frac{t^4}{t^2-1}\Delta t\).
9. \(\int\frac{t^2+2}{t^2-1}\Delta t\).
10. \(\int \frac{3t^4+3t^2+1}{t^2+1}\Delta t\).
11. \(\int \frac{t^5-t+3}{t^2-1}\nabla t\).
12. \( \int\frac{t^5-t+1}{t^2+1}\nabla t\).
13. \(\int\frac{-3t^4+3t^2+1}{t^2-1}\nabla t\).
14. \(\int \frac{1}{t+7}\Delta t\).
15. \( \int (t-3)^t \nabla t\).
16. \(\int \frac{1}{1+4t^2}\Delta t\).
17. \(\int \frac{1}{2-3t^2}\Delta t\).
18. \(\int \frac{1}{\sqrt{2-3t^2}}\Delta t\).
19. \(\int\frac{1}{3+4t^2}\nabla t\).
20. \(\int\frac{t^4+3t^2+4}{t^2+2}\Delta t\).
21. \(\int \frac{t^2+1}{t^4+1}\Delta t\).
22. \(\int \frac{t^2+1}{t\sqrt{t^4+1}}\Delta t\).
23. \(\int \frac{1}{t^2+6t+13}\nabla t\).
24. \(\int\frac{2t+11}{t^2+6t+13}\Delta t\).
25. \(\int\frac{t}{t^2+t+1}\nabla t\).
26. \( \int\frac{4t+8}{3t^2+2t+5}\nabla t\).
27. \(\int\frac{t^4}{t^2+3}\nabla t\).
28. \(\int\frac{t^3}{t^2+t+1}\Delta t\).
29. \(\int\frac{t^2+t+1}{t^2-t+1}\nabla t\).
30. \(\int\frac{t^3}{3t^4-2t^2+1}\nabla t\).
31. \(\int\frac{1}{\sqrt{3t^2-3t+5}}\Delta t\).
32. \( \int\frac{1}{\sqrt{3+t-t^2}}\nabla t\).
33. \(\int \frac{1}{t\sqrt{t^2-1}}\nabla t\).
34. \( \int \frac{1}{t\sqrt{t^2+t+1}}\nabla t\).
35. \(\int \frac{1}{t^2\sqrt{2t^2+2t+1}}\nabla t\).
36. \(\int \frac{1}{(1+t^2)^2}\nabla t\).
37. \( \int\frac{1}{(3+t^2)^2}\Delta t\).
38. \(\int\frac{1}{(3-2t)^4}\Delta t\).
39. \(\int \frac{1}{(1+t)(1+2t)(1+3t)}\Delta t\).
40. \(\int \frac{t}{(1+t)(2+t)^2}\Delta t\).
Let \(\mathbb{T}=[-1, 0]\cup \{2\}\cup 4^{\mathbb{N}}\). Find
1. \(e_{3, \mu}(t, -1)\).
2. \(\sin_{2, \mu}(t, -1)\).
3. \(\cos_{1, \mu}(t, 0)\)
4. \(cosh_{-1, \mu}(t, 1)\).
5. \(\sinh_{8, \mu}(t, -1)\).
6. \(h_{2, \mu}(t, 0)\).
7. \(h_{3, \mu}(t, 4)\).
8. \(e_{2, \mu}(t, -1)\ominus_\mu (h_{1, \mu}(t, -1)\oplus_\mu h_{3, \mu}(t, 0)\).
9. \(h_{2, \mu}(t, 0)\oplus_\mu h_{3, \mu}(t, 16)\).
10. \(\sin_{2, \mu}(t, 4)\oplus_\mu e_{-3, \mu}(t, 0)\).
11. \(e_{1, \nu}(t, 0)\).
12. \(\sin_{4, \nu}(t, -1)\).
13. \(\cos_{-1, \nu}(t, 1)\)
14. \(cosh_{-3, \nu}(t, 1)\).
15. \(\sinh_{1, \nu}(t, -1)\).
16. \(h_{3, \nu}(t, 1)\).
17. \(h_{2, \nu}(t, 0)\).
18. \(e_{2, \nu}(t, 0)\oplus_\nu (h_{1, \nu}(t, 2)\ominus_\nu h_{3, \nu}(t, 0)\).
19. \(h_{2, \nu}(t, 0)\oplus_\nu h_{3, \nu}(t, 16)\).
20. \(\sin_{2, \nu}(t, 4)\oplus_\nu e_{-3, \nu}(t, 0)\).
Let \(\mathbb{T}=2\mathbb{Z}\). Investigate for convergence the follwoing integrals.
1. \(\int\limits_1^\infty \frac{2t+1}{(t^2+1)(t^2+4t+_5)}\Delta t\).
2. \(\int\limits_2^\infty \frac{1}{4t^2+12t+5}\Delta t\).
3. \(\int\limits_0^\infty \frac{1}{t^2+10t+24}\Delta t\).
4. \(\int\limits_1^\infty \frac{1}{t^2+11t+25}\Delta t\).
5. \(\int\limits_1^\infty \frac{3t+7}{(t+1)(t+2)(t+3)(t+4)}\Delta t\).
6. \(\int\\limits_1^\infty \frac{\sin_{1, \mu}(t, 0)}{t^2}\Delta t\).
7. \(\int\limits_0^\infty \frac{1}{(2t+5)(3t+7)}\Delta t\).
8. \(\int\limits_1^\infty \frac{1}{t^3(t^2+5)(t^2+7t+1)}\nabla t\).
9. \(\int\limits_0^\infty \frac{t}{(t-1)(t+2)}\nabla t\).
10. \(\int\limits_0^\infty \frac{1}{2t+1}\nabla t\).
Let \(\mathbb{T}+3\mathbb{Z}\). Investigate for convergence the following series.
1. \(\sum\limits_{n=1}^\infty\left(\frac{1-t}{1+t}\right)^n\).
2. \(\sum\limits_{n=1}^\infty\frac{1}{(t+1)^n}\).
3. \(\sum\limits_{n=1}^\infty e_{-n, \mu}(t, 0)\).
4. \(\sum\limits_{n=1}^\infty (t^2+1)^n \).
5. \(\sum\limits_{n=1}^\infty\frac{1}{n^2t^2}\).
6. \(\sum\limits_{n=1}^\infty\frac{1}{(n+t)^3}\).
7. \(\sum\limits_{n=1}^\infty\frac{1}{2+n^2tn^4t^2}\).
8. \(\sum\limits_{n=1}^\infty\frac{nt}{1+n^2t^2}\).
9. \(\sum\limits_{n=1}^\infty\frac{t^n}{n!}\).
10. \(\sum\limits_{n=1}^\infty\frac{t^{n+1}}{n(n+1)}\).
11. \(\sum\limits_{n=1}^\infty 3^{n^2}t^{n^3}\).
12. \(\sum\limits_{n=1}^\infty \frac{t^n}{(n!)^\alpha}\).
13. \(\sum\limits_{n=1}^\infty \frac{(3t)^n}{n\sqrt{n+t}}\).
14. \(\sum\limits_{n=1}^\infty\frac{n}{(t-1)^n}\).
15. \(\sum\limits_{n=1}^\infty\frac{1+nt}{(3-nt)^4}\).
16. \(\sum\limits_{n=1}^\infty\frac{2-n^3t^3}{1+n^6t^6}\).
17. \(\sum\limits_{n=1}^\infty\frac{n-t}{(n+t)^4}\).
18. \(\sum\limits_{n=1}^\infty\frac{1}{(1+nt+n^4t^4)^5}\).
19. \(\sum\limits_{n=1}^\infty\frac{1}{\sqrt{n^2+t^2}}\).
20. \(\sum\limits_{n=1}^\infty\frac{4^n}{n^2t^8+2}\).