## Problem Set: Power Series and Functions

In the following exercises, state whether each statement is true, or give an example to show that it is false.

1. If $\displaystyle\sum _{n=1}^{\infty }{a}_{n}{x}^{n}$ converges, then ${a}_{n}{x}^{n}\to 0$ as $n\to \infty$.

2. $\displaystyle\sum _{n=1}^{\infty }{a}_{n}{x}^{n}$ converges at $x=0$ for any real numbers ${a}_{n}$.

3. Given any sequence ${a}_{n}$, there is always some $R>0$, possibly very small, such that $\displaystyle\sum _{n=1}^{\infty }{a}_{n}{x}^{n}$ converges on $\left(\text{-}R,R\right)$.

4. If $\displaystyle\sum _{n=1}^{\infty }{a}_{n}{x}^{n}$ has radius of convergence $R>0$ and if $|{b}_{n}|\le |{a}_{n}|$ for all n, then the radius of convergence of $\displaystyle\sum _{n=1}^{\infty }{b}_{n}{x}^{n}$ is greater than or equal to R.

5. Suppose that $\displaystyle\sum _{n=0}^{\infty }{a}_{n}{\left(x - 3\right)}^{n}$ converges at $x=6$. At which of the following points must the series also converge? Use the fact that if $\displaystyle\sum {a}_{n}{\left(x-c\right)}^{n}$ converges at x, then it converges at any point closer to c than x.

1. $x=1$
2. $x=2$
3. $x=3$
4. $x=0$
5. $x=5.99$
6. $x=0.000001$

6. Suppose that $\displaystyle\sum _{n=0}^{\infty }{a}_{n}{\left(x+1\right)}^{n}$ converges at $x=-2$.
At which of the following points must the series also converge? Use the fact that if $\displaystyle\sum {a}_{n}{\left(x-c\right)}^{n}$ converges at x, then it converges at any point closer to c than x.

1. $x=2$
2. $x=-1$
3. $x=-3$
4. $x=0$
5. $x=0.99$
6. $x=0.000001$

In the following exercises, suppose that $|\frac{{a}_{n+1}}{{a}_{n}}|\to 1$ as $n\to \infty$. Find the radius of convergence for each series.

7. $\displaystyle\sum _{n=0}^{\infty }{a}_{n}{2}^{n}{x}^{n}$

8. $\displaystyle\sum _{n=0}^{\infty }\frac{{a}_{n}{x}^{n}}{{2}^{n}}$

9. $\displaystyle\sum _{n=0}^{\infty }\frac{{a}_{n}{\pi }^{n}{x}^{n}}{{e}^{n}}$

10. $\displaystyle\sum _{n=0}^{\infty }\frac{{a}_{n}{\left(-1\right)}^{n}{x}^{n}}{{10}^{n}}$

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

12. $\displaystyle\sum _{n=0}^{\infty }{a}_{n}{\left(-4\right)}^{n}{x}^{2n}$

In the following exercises, find the radius of convergence R and interval of convergence for $\displaystyle\sum {a}_{n}{x}^{n}$ with the given coefficients ${a}_{n}$.

13. $\displaystyle\sum _{n=1}^{\infty }\frac{{\left(2x\right)}^{n}}{n}$

14. $\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n}\frac{{x}^{n}}{\sqrt{n}}$

15. $\displaystyle\sum _{n=1}^{\infty }\frac{n{x}^{n}}{{2}^{n}}$

16. $\displaystyle\sum _{n=1}^{\infty }\frac{n{x}^{n}}{{e}^{n}}$

17. $\displaystyle\sum _{n=1}^{\infty }\frac{{n}^{2}{x}^{n}}{{2}^{n}}$

18. $\displaystyle\sum _{k=1}^{\infty }\frac{{k}^{e}{x}^{k}}{{e}^{k}}$

19. $\displaystyle\sum _{k=1}^{\infty }\frac{{\pi }^{k}{x}^{k}}{{k}^{\pi }}$

20. $\displaystyle\sum _{n=1}^{\infty }\frac{{x}^{n}}{n\text{!}}$

21. $\displaystyle\sum _{n=1}^{\infty }\frac{{10}^{n}{x}^{n}}{n\text{!}}$

22. $\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n}\frac{{x}^{n}}{\text{ln}\left(2n\right)}$

In the following exercises, find the radius of convergence of each series.

23. $\displaystyle\sum _{k=1}^{\infty }\frac{{\left(k\text{!}\right)}^{2}{x}^{k}}{\left(2k\right)\text{!}}$

24. $\displaystyle\sum _{n=1}^{\infty }\frac{\left(2n\right)\text{!}{x}^{n}}{{n}^{2n}}$

25. $\displaystyle\sum _{k=1}^{\infty }\frac{k\text{!}}{1\cdot 3\cdot 5\text{\cdots }\left(2k - 1\right)}{x}^{k}$

26. $\displaystyle\sum _{k=1}^{\infty }\frac{2\cdot 4\cdot 6\text{\cdots }2k}{\left(2k\right)\text{!}}{x}^{k}$

27. $\displaystyle\sum _{n=1}^{\infty }\frac{{x}^{n}}{\left(\begin{array}{c}2n\\ n\end{array}\right)}$ where $\left(\begin{array}{c}n\\ k\end{array}\right)=\frac{n\text{!}}{k\text{!}\left(n-k\right)\text{!}}$

28. $\displaystyle\sum _{n=1}^{\infty }{\sin}^{2}n{x}^{n}$

In the following exercises, use the ratio test to determine the radius of convergence of each series.

29. $\displaystyle\sum _{n=1}^{\infty }\frac{{\left(n\text{!}\right)}^{3}}{\left(3n\right)\text{!}}{x}^{n}$

30. $\displaystyle\sum _{n=1}^{\infty }\frac{{2}^{3n}{\left(n\text{!}\right)}^{3}}{\left(3n\right)\text{!}}{x}^{n}$

31. $\displaystyle\sum _{n=1}^{\infty }\frac{n\text{!}}{{n}^{n}}{x}^{n}$

32. $\displaystyle\sum _{n=1}^{\infty }\frac{\left(2n\right)\text{!}}{{n}^{2n}}{x}^{n}$

In the following exercises, given that $\frac{1}{1-x}=\displaystyle\sum _{n=0}^{\infty }{x}^{n}$ with convergence in $\left(-1,1\right)$, find the power series for each function with the given center a, and identify its interval of convergence.

33. $f\left(x\right)=\frac{1}{x};a=1$ (Hint: $\frac{1}{x}=\frac{1}{1-\left(1-x\right)}$)

34. $f\left(x\right)=\frac{1}{1-{x}^{2}};a=0$

35. $f\left(x\right)=\frac{x}{1-{x}^{2}};a=0$

36. $f\left(x\right)=\frac{1}{1+{x}^{2}};a=0$

37. $f\left(x\right)=\frac{{x}^{2}}{1+{x}^{2}};a=0$

38. $f\left(x\right)=\frac{1}{2-x};a=1$

39. $f\left(x\right)=\frac{1}{1 - 2x};a=0$.

40. $f\left(x\right)=\frac{1}{1 - 4{x}^{2}};a=0$

41. $f\left(x\right)=\frac{{x}^{2}}{1 - 4{x}^{2}};a=0$

42. $f\left(x\right)=\frac{{x}^{2}}{5 - 4x+{x}^{2}};a=2$

Use the next exercise to find the radius of convergence of the given series in the subsequent exercises.

43. Explain why, if ${|{a}_{n}|}^{\frac{1}{n}}\to r>0$, then ${|{a}_{n}{x}^{n}|}^{\frac{1}{n}}\to |x|r<1$ whenever $|x|<\frac{1}{r}$ and, therefore, the radius of convergence of $\displaystyle\sum _{n=1}^{\infty }{a}_{n}{x}^{n}$ is $R=\frac{1}{r}$.

44. $\displaystyle\sum _{n=1}^{\infty }\frac{{x}^{n}}{{n}^{n}}$

45. $\displaystyle\sum _{k=1}^{\infty }{\left(\frac{k - 1}{2k+3}\right)}^{k}{x}^{k}$

46. $\displaystyle\sum _{k=1}^{\infty }{\left(\frac{2{k}^{2}-1}{{k}^{2}+3}\right)}^{k}{x}^{k}$

47. $\displaystyle\sum _{n=1}^{\infty }{a}_{n}={\left({n}^{\frac{1}{n}}-1\right)}^{n}{x}^{n}$

48. Suppose that $p\left(x\right)=\displaystyle\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$ such that ${a}_{n}=0$ if n is odd. Explain why $p\left(x\right)=-p\left(\text{-}x\right)$.

49. Suppose that $p\left(x\right)=\displaystyle\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$ such that ${a}_{n}=0$ if n is even. Explain why $p\left(x\right)=p\left(\text{-}x\right)$.

50. Suppose that $p\left(x\right)=\displaystyle\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$ converges on $\left(-1,1\right]$.
Find the interval of convergence of $p\left(Ax\right)$.

51. Suppose that $p\left(x\right)=\displaystyle\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$ converges on $\left(-1,1\right]$. Find the interval of convergence of $p\left(2x - 1\right)$.

In the following exercises, suppose that $p\left(x\right)=\displaystyle\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$ satisfies $\underset{n\to \infty }{\text{lim}}\frac{{a}_{n+1}}{{a}_{n}}=1$ where ${a}_{n}\ge 0$ for each n. State whether each series converges on the full interval $\left(-1,1\right)$, or if there is not enough information to draw a conclusion. Use the comparison test when appropriate.

52. $\displaystyle\sum _{n=0}^{\infty }{a}_{n}{x}^{2n}$

53. $\displaystyle\sum _{n=0}^{\infty }{a}_{2n}{x}^{2n}$

54. $\displaystyle\sum _{n=0}^{\infty }{a}_{2n}{x}^{n}\left(Hint\text{:}x=\pm \sqrt{{x}^{2}}\right)$

55. $\displaystyle\sum _{n=0}^{\infty }{a}_{{n}^{2}}{x}^{{n}^{2}}$ (Hint: Let ${b}_{k}={a}_{k}$ if $k={n}^{2}$ for some n, otherwise ${b}_{k}=0.$)

56. Suppose that $p\left(x\right)$ is a polynomial of degree N. Find the radius and interval of convergence of $\displaystyle\sum _{n=1}^{\infty }p\left(n\right){x}^{n}$.

57. [T] Plot the graphs of $\frac{1}{1-x}$ and of the partial sums ${S}_{N}=\displaystyle\sum _{n=0}^{N}{x}^{n}$ for $n=10,20,30$ on the interval $\left[-0.99,0.99\right]$. Comment on the approximation of $\frac{1}{1-x}$ by ${S}_{N}$ near $x=-1$ and near $x=1$ as N increases.

58. [T] Plot the graphs of $\text{-}\text{ln}\left(1-x\right)$ and of the partial sums ${S}_{N}=\displaystyle\sum _{n=1}^{N}\frac{{x}^{n}}{n}$ for $n=10,50,100$ on the interval $\left[-0.99,0.99\right]$. Comment on the behavior of the sums near $x=-1$ and near $x=1$ as N increases.

59. [T] Plot the graphs of the partial sums ${S}_{n}=\displaystyle\sum _{n=1}^{N}\frac{{x}^{n}}{{n}^{2}}$ for $n=10,50,100$ on the interval $\left[-0.99,0.99\right]$. Comment on the behavior of the sums near $x=-1$ and near $x=1$ as N increases.

60. [T] Plot the graphs of the partial sums ${S}_{N}=\displaystyle\sum _{n=1}^{N}\sin{n}{x}^{n}$ for $n=10,50,100$ on the interval $\left[-0.99,0.99\right]$. Comment on the behavior of the sums near $x=-1$ and near $x=1$ as N increases.

61. [T] Plot the graphs of the partial sums ${S}_{N}=\displaystyle\sum _{n=0}^{N}{\left(-1\right)}^{n}\frac{{x}^{2n+1}}{\left(2n+1\right)\text{!}}$ for $n=3,5,10$ on the interval $\left[-2\pi ,2\pi \right]$. Comment on how these plots approximate $\sin{x}$ as N increases.

62. [T] Plot the graphs of the partial sums ${S}_{N}=\displaystyle\sum _{n=0}^{N}{\left(-1\right)}^{n}\frac{{x}^{2n}}{\left(2n\right)\text{!}}$ for $n=3,5,10$ on the interval $\left[-2\pi ,2\pi \right]$. Comment on how these plots approximate $\cos{x}$ as N increases.