## Module 6 Review Problems

True or False? In the following exercises, justify your answer with a proof or a counterexample.

1. If the radius of convergence for a power series $\displaystyle\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$ is $5$, then the radius of convergence for the series $\displaystyle\sum _{n=1}^{\infty }n{a}_{n}{x}^{n - 1}$ is also $5$.

2. Power series can be used to show that the derivative of ${e}^{x}\text{ is }{e}^{x}$. (Hint: Recall that ${e}^{x}=\displaystyle\sum _{n=0}^{\infty }\frac{1}{n\text{!}}{x}^{n}.$)

3. For small values of $x,\sin{x}\approx x$.

4. The radius of convergence for the Maclaurin series of $f\left(x\right)={3}^{x}$ is $3$.

In the following exercises, find the radius of convergence and the interval of convergence for the given series.

5. $\displaystyle\sum _{n=0}^{\infty }{n}^{2}{\left(x - 1\right)}^{n}$

6. $\displaystyle\sum _{n=0}^{\infty }\frac{{x}^{n}}{{n}^{n}}$

7. $\displaystyle\sum _{n=0}^{\infty }\frac{3n{x}^{n}}{{12}^{n}}$

8. $\displaystyle\sum _{n=0}^{\infty }\frac{{2}^{n}}{{e}^{n}}{\left(x-e\right)}^{n}$

In the following exercises, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series.

9. $f\left(x\right)=\frac{{x}^{2}}{x+3}$

10. $f\left(x\right)=\frac{8x+2}{2{x}^{2}-3x+1}$

In the following exercises, find the power series for the given function using term-by-term differentiation or integration.

11. $f\left(x\right)={\tan}^{-1}\left(2x\right)$

12. $f\left(x\right)=\frac{x}{{\left(2+{x}^{2}\right)}^{2}}$

In the following exercises, evaluate the Taylor series expansion of degree four for the given function at the specified point. What is the error in the approximation?

13. $f\left(x\right)={x}^{3}-2{x}^{2}+4,a=-3$

14. $f\left(x\right)={e}^{\frac{1}{\left(4x\right)}},a=4$

In the following exercises, find the Maclaurin series for the given function.

15. $f\left(x\right)=\cos\left(3x\right)$

16. $f\left(x\right)=\text{ln}\left(x+1\right)$

In the following exercises, find the Taylor series at the given value.

17. $f\left(x\right)=\sin{x},a=\frac{\pi }{2}$

18. $f\left(x\right)=\frac{3}{x},a=1$

In the following exercises, find the Maclaurin series for the given function.

19. $f\left(x\right)={e}^{\text{-}{x}^{2}}-1$

20. $f\left(x\right)=\cos{x}-x\sin{x}$

In the following exercises, find the Maclaurin series for $F\left(x\right)={\displaystyle\int }_{0}^{x}f\left(t\right)dt$ by integrating the Maclaurin series of $f\left(x\right)$ term by term.

21. $f\left(x\right)=\frac{\sin{x}}{x}$

22. $f\left(x\right)=1-{e}^{x}$

23. Use power series to prove Euler’s formula: ${e}^{ix}=\cos{x}+i\sin{x}$

The following exercises 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 $1\text{%},5\text{%},\text{and }10\text{%}$.

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 $250,000?$

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 $1\text{%},5\text{%},\text{and }10\text{%}$.