10.R: Chapter 10 Review Exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
True or False? In exercises 1 - 4, justify your answer with a proof or a counterexample.
1) If the radius of convergence for a power series \displaystyle \sum_{n=0}^∞a_nx^n is 5, then the radius of convergence for the series \displaystyle \sum_{n=1}^∞na_nx^{n−1} is also 5.
- Answer
- True
2) Power series can be used to show that the derivative of e^x is e^x. (Hint: Recall that \displaystyle e^x=\sum_{n=0}^∞\frac{1}{n!}x^n.)
3) For small values of x, \sin x ≈ x.
- Answer
- True
4) The radius of convergence for the Maclaurin series of f(x)=3^x is 3.
In exercises 5 - 8, find the radius of convergence and the interval of convergence for the given series.
5) \displaystyle \sum_{n=0}^∞n^2(x−1)^n
- Answer
- ROC: 1; IOC: (0,2)
6) \displaystyle \sum_{n=0}^∞\frac{x^n}{n^n}
7) \displaystyle \sum_{n=0}^∞\frac{3nx^n}{12^n}
- Answer
- ROC: 12; IOC: (−16,8)
8) \displaystyle \sum_{n=0}^∞\frac{2^n}{e^n}(x−e)^n
In exercises 9 - 10, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series.
9) f(x)=\dfrac{x^2}{x+3}
- Answer
- \displaystyle \sum_{n=0}^∞\frac{(−1)^n}{3^{n+1}}x^n; ROC: 3; IOC: (−3,3)
10) f(x)=\dfrac{8x+2}{2x^2−3x+1}
In exercises 11 - 12, find the power series for the given function using term-by-term differentiation or integration.
11) f(x)=\tan^{−1}(2x)
- Answer
- integration: \displaystyle \sum_{n=0}^∞\frac{(−1)^n}{2n+1}(2x)^{2n+1}
12) f(x)=\dfrac{x}{(2+x^2)^2}
In exercises 13 - 14, evaluate the Taylor series expansion of degree four for the given function at the specified point. What is the error in the approximation?
13) f(x)=x^3−2x^2+4, \quad a=−3
- Answer
- p_4(x)=(x+3)^3−11(x+3)^2+39(x+3)−41; exact
14) f(x)=e^{1/(4x)}, \quad a=4
In exercises 15 - 16, find the Maclaurin series for the given function.
15) f(x)=\cos(3x)
- Answer
- \displaystyle \sum_{n=0}^∞\frac{(−1)^n(3x)^{2n}}{2n!}
16) f(x)=\ln(x+1)
In exercises 17 - 18, find the Taylor series at the given value.
17) f(x)=\sin x, \quad a=\frac{π}{2}
- Answer
- \displaystyle \sum_{n=0}^∞\frac{(−1)^n}{(2n)!}\left(x−\frac{π}{2}\right)^{2n}
18) f(x)=\dfrac{3}{x},\quad a=1
In exercises 19 - 20, find the Maclaurin series for the given function.
19) f(x)=e^{−x^2}−1
- Answer
- \displaystyle \sum_{n=1}^∞\frac{(−1)^n}{n!}x^{2n}
20) f(x)=\cos x−x\sin x
In exercises 21 - 23, find the Maclaurin series for \displaystyle F(x)=∫^x_0f(t)\,dt by integrating the Maclaurin series of f(x) term by term.
21) f(x)=\dfrac{\sin x}{x}
- Answer
- \displaystyle F(x)=\sum_{n=0}^∞\frac{(−1)^n}{(2n+1)(2n+1)!}x^{2n+1}
22) f(x)=1−e^x
23) Use power series to prove Euler’s formula: e^{ix}=cosx+isinx
- Answer
- Answers may vary.
Exercises 24 - 26 consider problems of annuity payments.
24) For annuities with a present value of $1 million, calculate the annual payouts given over 25 years assuming interest rates of 1\%,5\%, and 10\%.
25) A lottery winner has an annuity that has a present value of $10 million. What interest rate would they need to live on perpetual annual payments of $250,000?
- Answer
- 2.5\%
26) Calculate the necessary present value of an annuity in order to support annual payouts of $15,000 given over 25 years assuming interest rates of 1\%,5\%,and 10\%.