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 $5$ .

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2. Power series can be used to show that the derivative of

e^{x} is e^{x}

${e}^{x}\text{ is }{e}^{x}$ . (Hint: Recall that

e^{x} = \infty \sum n = 0 \frac{1}{n!} x^{n} .

${e}^{x}=\displaystyle\sum _{n=0}^{\infty }\frac{1}{n\text{!}}{x}^{n}.$ )

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

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4. The radius of convergence for the Maclaurin series of

f (x) = 3^{x}

$f\left(x\right)={3}^{x}$ is

3

$3$ .

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

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

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ROC:

1

$1$ ; IOC:

(0, 2)

$\left(0,2\right)$

\infty \sum n = 0 \frac{x^{n}}{n^{n}}

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

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

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ROC:

12

$12$ ; IOC:

(- 16, 8)

$\left(-16,8\right)$

\infty \sum n = 0 \frac{2^{n}}{e^{n}} {(x - e)}^{n}

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

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

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\infty \sum n = 0 \frac{{(- 1)}^{n}}{3^{n + 1}} x^{n}

$\displaystyle\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}}{{3}^{n+1}}{x}^{n}$ ; ROC:

3

$3$ ; IOC:

(- 3, 3)

$\left(-3,3\right)$

10.

f (x) = \frac{8 x + 2}{2 x^{2} - 3 x + 1}

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

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

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

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integration:

\infty \sum n = 0 \frac{{(- 1)}^{n}}{2 n + 1} {(2 x)}^{2 n + 1}

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

12.

f (x) = \frac{x}{{(2 + x^{2})}^{2}}

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

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

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p_{4} (x) = {(x + 3)}^{3} - 11 {(x + 3)}^{2} + 39 (x + 3) - 41

${p}_{4}\left(x\right)={\left(x+3\right)}^{3}-11{\left(x+3\right)}^{2}+39\left(x+3\right)-41$ ; exact

14.

f (x) = e^{\frac{1}{(4 x)}}, a = 4

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

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

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

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\infty \sum n = 0 \frac{{(- 1)}^{n} {(3 x)}^{2 n}}{2 n!}

$\displaystyle\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}{\left(3x\right)}^{2n}}{2n\text{!}}$

16.

f (x) = ln (x + 1)

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

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

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

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\infty \sum n = 0 \frac{{(- 1)}^{n}}{(2 n)!} {(x - \frac{π}{2})}^{2 n}

$\displaystyle\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}}{\left(2n\right)\text{!}}{\left(x-\frac{\pi }{2}\right)}^{2n}$

18.

f (x) = \frac{3}{x}, a = 1

$f\left(x\right)=\frac{3}{x},a=1$

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

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

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\infty \sum n = 1 \frac{{(- 1)}^{n}}{n!} x^{2 n}

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

20.

f (x) = cos x - x sin x

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

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

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$F\left(x\right)=\displaystyle\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}}{\left(2n+1\right)\left(2n+1\right)\text{!}}{x}^{2n+1}$

22.

$f\left(x\right)=1-{e}^{x}$

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

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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?$

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$2.5\text{%}$

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

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

Power Series: Problem Sets

Module 6 Review Problems

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