## Problem Set: Properties of Power Series

1. If $f\left(x\right)=\displaystyle\sum _{n=0}^{\infty }\frac{{x}^{n}}{n\text{!}}$ and $g\left(x\right)=\displaystyle\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{x}^{n}}{n\text{!}}$, find the power series of $\frac{1}{2}\left(f\left(x\right)+g\left(x\right)\right)$ and of $\frac{1}{2}\left(f\left(x\right)-g\left(x\right)\right)$.

2. If $C\left(x\right)=\displaystyle\sum _{n=0}^{\infty }\frac{{x}^{2n}}{\left(2n\right)\text{!}}$ and $S\left(x\right)=\displaystyle\sum _{n=0}^{\infty }\frac{{x}^{2n+1}}{\left(2n+1\right)\text{!}}$, find the power series of $C\left(x\right)+S\left(x\right)$ and of $C\left(x\right)-S\left(x\right)$.

In the following exercises, use partial fractions to find the power series of each function.

3. $\frac{4}{\left(x - 3\right)\left(x+1\right)}$

4. $\frac{3}{\left(x+2\right)\left(x - 1\right)}$

5. $\frac{5}{\left({x}^{2}+4\right)\left({x}^{2}-1\right)}$

6. $\frac{30}{\left({x}^{2}+1\right)\left({x}^{2}-9\right)}$

In the following exercises, express each series as a rational function.

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

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

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

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

The following exercises explore applications of annuities.

11. Calculate the present values P of an annuity in which $10,000 is to be paid out annually for a period of 20 years, assuming interest rates of $r=0.03,r=0.05$, and $r=0.07$. 12. Calculate the present values P of annuities in which$9,000 is to be paid out annually perpetually, assuming interest rates of $r=0.03,r=0.05$ and $r=0.07$.

13. Calculate the annual payouts C to be given for 20 years on annuities having present value $100,000 assuming respective interest rates of $r=0.03,r=0.05$, and $r=0.07$. 14. Calculate the annual payouts C to be given perpetually on annuities having present value$100,000 assuming respective interest rates of $r=0.03,r=0.05$, and $r=0.07$.

15. Suppose that an annuity has a present value $P=1\text{million dollars}$. What interest rate r would allow for perpetual annual payouts of $50,000? 16. Suppose that an annuity has a present value $P=10\text{million dollars}\text{.}$ What interest rate r would allow for perpetual annual payouts of$100,000?

In the following exercises, express the sum of each power series in terms of geometric series, and then express the sum as a rational function.

17. $x+{x}^{2}-{x}^{3}+{x}^{4}+{x}^{5}-{x}^{6}+\cdots$ (Hint: Group powers x3k, ${x}^{3k - 1}$, and ${x}^{3k - 2}.$)

18. $x+{x}^{2}-{x}^{3}-{x}^{4}+{x}^{5}+{x}^{6}-{x}^{7}-{x}^{8}+\cdots$ (Hint: Group powers x4k, ${x}^{4k - 1}$, etc.)

19. $x-{x}^{2}-{x}^{3}+{x}^{4}-{x}^{5}-{x}^{6}+{x}^{7}-\cdots$ (Hint: Group powers x3k, ${x}^{3k - 1}$, and ${x}^{3k - 2}.$)

20. $\frac{x}{2}+\frac{{x}^{2}}{4}-\frac{{x}^{3}}{8}+\frac{{x}^{4}}{16}+\frac{{x}^{5}}{32}-\frac{{x}^{6}}{64}+\cdots$ (Hint: Group powers ${\left(\frac{x}{2}\right)}^{3k},{\left(\frac{x}{2}\right)}^{3k - 1}$, and ${\left(\frac{x}{2}\right)}^{3k - 2}.$)

In the following exercises, find the power series of $f\left(x\right)g\left(x\right)$ given f and g as defined.

21. $f\left(x\right)=2\displaystyle\sum _{n=0}^{\infty }{x}^{n},g\left(x\right)=\displaystyle\sum _{n=0}^{\infty }n{x}^{n}$

22. $f\left(x\right)=\displaystyle\sum _{n=1}^{\infty }{x}^{n},g\left(x\right)=\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}{x}^{n}$. Express the coefficients of $f\left(x\right)g\left(x\right)$ in terms of ${H}_{n}=\displaystyle\sum _{k=1}^{n}\frac{1}{k}$.

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

24. $f\left(x\right)=g\left(x\right)=\displaystyle\sum _{n=1}^{\infty }n{x}^{n}$

In the following exercises, differentiate the given series expansion of f term-by-term to obtain the corresponding series expansion for the derivative of f.

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

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

In the following exercises, integrate the given series expansion of $f$ term-by-term from zero to x to obtain the corresponding series expansion for the indefinite integral of $f$.

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

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

In the following exercises, evaluate each infinite series by identifying it as the value of a derivative or integral of geometric series.

29. Evaluate $\displaystyle\sum _{n=1}^{\infty }\frac{n}{{2}^{n}}$ as ${f}^{\prime }\left(\frac{1}{2}\right)$ where $f\left(x\right)=\displaystyle\sum _{n=0}^{\infty }{x}^{n}$.

30. Evaluate $\displaystyle\sum _{n=1}^{\infty }\frac{n}{{3}^{n}}$ as ${f}^{\prime }\left(\frac{1}{3}\right)$ where $f\left(x\right)=\displaystyle\sum _{n=0}^{\infty }{x}^{n}$.

31. Evaluate $\displaystyle\sum _{n=2}^{\infty }\frac{n\left(n - 1\right)}{{2}^{n}}$ as $f''\left(\frac{1}{2}\right)$ where $f\left(x\right)=\displaystyle\sum _{n=0}^{\infty }{x}^{n}$.

32. Evaluate $\displaystyle\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}}{n+1}$ as ${\displaystyle\int }_{0}^{1}f\left(t\right)dt$ where $f\left(x\right)=\displaystyle\sum _{n=0}^{\infty }{\left(-1\right)}^{n}{x}^{2n}=\frac{1}{1+{x}^{2}}$.

In the following exercises, given that $\frac{1}{1-x}=\displaystyle\sum _{n=0}^{\infty }{x}^{n}$, use term-by-term differentiation or integration to find power series for each function centered at the given point.

33. $f\left(x\right)=\text{ln}x$ centered at $x=1$ (Hint: $x=1-\left(1-x\right)$)

34. $\text{ln}\left(1-x\right)$ at $x=0$

35. $\text{ln}\left(1-{x}^{2}\right)$ at $x=0$

36. $f\left(x\right)=\frac{2x}{{\left(1-{x}^{2}\right)}^{2}}$ at $x=0$

37. $f\left(x\right)={\tan}^{-1}\left({x}^{2}\right)$ at $x=0$

38. $f\left(x\right)=\text{ln}\left(1+{x}^{2}\right)$ at $x=0$

39. $f\left(x\right)={\displaystyle\int }_{0}^{x}\text{ln}tdt$ where $\text{ln}\left(x\right)=\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n - 1}\frac{{\left(x - 1\right)}^{n}}{n}$

40. [T] Evaluate the power series expansion $\text{ln}\left(1+x\right)=\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n - 1}\frac{{x}^{n}}{n}$ at $x=1$ to show that $\text{ln}\left(2\right)$ is the sum of the alternating harmonic series. Use the alternating series test to determine how many terms of the sum are needed to estimate $\text{ln}\left(2\right)$ accurate to within 0.001, and find such an approximation.

41. [T] Subtract the infinite series of $\text{ln}\left(1-x\right)$ from $\text{ln}\left(1+x\right)$ to get a power series for $\text{ln}\left(\frac{1+x}{1-x}\right)$. Evaluate at $x=\frac{1}{3}$. What is the smallest N such that the Nth partial sum of this series approximates $\text{ln}\left(2\right)$ with an error less than 0.001?

In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum.

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

43. $\displaystyle\sum _{k=1}^{\infty }\frac{{x}^{3k}}{6k}$

44. $\displaystyle\sum _{k=1}^{\infty }{\left(1+{x}^{2}\right)}^{\text{-}k}$ using $y=\frac{1}{1+{x}^{2}}$

45. $\displaystyle\sum _{k=1}^{\infty }{2}^{\text{-}kx}$ using $y={2}^{\text{-}x}$

46. Show that, up to powers x3 and y3, $E\left(x\right)=\displaystyle\sum _{n=0}^{\infty }\frac{{x}^{n}}{n\text{!}}$ satisfies $E\left(x+y\right)=E\left(x\right)E\left(y\right)$.

47. Differentiate the series $E\left(x\right)=\displaystyle\sum _{n=0}^{\infty }\frac{{x}^{n}}{n\text{!}}$ term-by-term to show that $E\left(x\right)$ is equal to its derivative.

48. Show that if $f\left(x\right)=\displaystyle\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$ is a sum of even powers, that is, ${a}_{n}=0$ if n is odd, then $F={\displaystyle\int }_{0}^{x}f\left(t\right)dt$ is a sum of odd powers, while if f is a sum of odd powers, then F is a sum of even powers.

49. [T] Suppose that the coefficients an of the series $\displaystyle\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$ are defined by the recurrence relation ${a}_{n}=\frac{{a}_{n - 1}}{n}+\frac{{a}_{n - 2}}{n\left(n - 1\right)}$. For ${a}_{0}=0$ and ${a}_{1}=1$, compute and plot the sums ${S}_{N}=\displaystyle\sum _{n=0}^{N}{a}_{n}{x}^{n}$ for $N=2,3,4,5$ on $\left[-1,1\right]$.

50. [T] Suppose that the coefficients an of the series $\displaystyle\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$ are defined by the recurrence relation ${a}_{n}=\frac{{a}_{n - 1}}{\sqrt{n}}-\frac{{a}_{n - 2}}{\sqrt{n\left(n - 1\right)}}$. For ${a}_{0}=1$ and ${a}_{1}=0$, compute and plot the sums ${S}_{N}=\displaystyle\sum _{n=0}^{N}{a}_{n}{x}^{n}$ for $N=2,3,4,5$ on $\left[-1,1\right]$.

51. [T] Given the power series expansion $\text{ln}\left(1+x\right)=\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n - 1}\frac{{x}^{n}}{n}$, determine how many terms N of the sum evaluated at $x=-\frac{1}{2}$ are needed to approximate $\text{ln}\left(2\right)$ accurate to within $\frac{1}{1000}$. Evaluate the corresponding partial sum $\displaystyle\sum _{n=1}^{N}{\left(-1\right)}^{n - 1}\frac{{x}^{n}}{n}$.

52. [T] Given the power series expansion ${\tan}^{-1}\left(x\right)=\displaystyle\sum _{k=0}^{\infty }{\left(-1\right)}^{k}\frac{{x}^{2k+1}}{2k+1}$, use the alternating series test to determine how many terms N of the sum evaluated at $x=1$ are needed to approximate ${\tan}^{-1}\left(1\right)=\frac{\pi }{4}$ accurate to within $\frac{1}{1000}$. Evaluate the corresponding partial sum $\displaystyle\sum _{k=0}^{N}{\left(-1\right)}^{k}\frac{{x}^{2k+1}}{2k+1}$.

53. [T] Recall that ${\tan}^{-1}\left(\frac{1}{\sqrt{3}}\right)=\frac{\pi }{6}$. Assuming an exact value of $\left(\frac{1}{\sqrt{3}}\right)$, estimate $\frac{\pi }{6}$ by evaluating partial sums ${S}_{N}\left(\frac{1}{\sqrt{3}}\right)$ of the power series expansion ${\tan}^{-1}\left(x\right)=\displaystyle\sum _{k=0}^{\infty }{\left(-1\right)}^{k}\frac{{x}^{2k+1}}{2k+1}$ at $x=\frac{1}{\sqrt{3}}$. What is the smallest number N such that $6{S}_{N}\left(\frac{1}{\sqrt{3}}\right)$ approximates π accurately to within 0.001? How many terms are needed for accuracy to within 0.00001?