2.6: Chapter 2 Review Exercises

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 $10\%.$