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# 1 E: Chapter Exercises

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## Exercise $$\PageIndex{1}$$

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 $$\displaystyle \sum_{n=0}^∞a_nx^n$$ is $$\displaystyle 5$$, then the radius of convergence for the series $$\displaystyle \sum_{n=1}^∞na_nx^{n−1}$$ is also $$\displaystyle 5$$.

True

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

3. For small values of $$\displaystyle x,sinx≈x.$$

True

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

## Exercise $$\PageIndex{2}$$

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

1. $$\displaystyle \sum_{n=0}^∞n^2(x−1)^n$$

ROC: $$\displaystyle 1$$; IOC: $$\displaystyle (0,2)$$

2. $$\displaystyle \sum_{n=0}^∞\frac{x^n}{n^n}$$

3. $$\displaystyle \sum_{n=0}^∞\frac{3nx^n}{12^n}$$

ROC: $$\displaystyle 12;$$ IOC: $$\displaystyle (−16,8)$$

4. $$\displaystyle \sum_{n=0}^∞\frac{2^n}{e^n}(x−e)^n$$

## Exercise $$\PageIndex{3}$$

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.

1. $$\displaystyle f(x)=\frac{x^2}{x+3}$$

$$\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{3^{n+1}}x^n;$$ ROC: $$\displaystyle 3$$; IOC: $$\displaystyle (−3,3)$$

2. $$\displaystyle f(x)=\frac{8x+2}{2x^2−3x+1}$$

## Exercise $$\PageIndex{4}$$

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

1. $$\displaystyle f(x)=tan^{−1}(2x)$$

integration: $$\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{2n+1}(2x)^{2n+1}$$

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

## Exercise $$\PageIndex{5}$$

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?

1. $$\displaystyle f(x)=x^3−2x^2+4,a=−3$$

$$\displaystyle p_4(x)=(x+3)^3−11(x+3)^2+39(x+3)−41;$$ exact

2. $$\displaystyle f(x)=e^{1/(4x)},a=4$$

## Exercise $$\PageIndex{6}$$

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

1. $$\displaystyle f(x)=cos(3x)$$

$$\displaystyle \sum_{n=0}^∞\frac{(−1)^n(3x)^{2n}}{2n!}$$

2. $$\displaystyle f(x)=ln(x+1)$$

## Exercise $$\PageIndex{7}$$

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

1. $$\displaystyle f(x)=sinx,a=\frac{π}{2}$$

$$\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{(2n)!}(x−\frac{π}{2})^{2n}$$

2. $$\displaystyle f(x)=\frac{3}{x},a=1$$

## Exercise $$\PageIndex{8}$$

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

1. $$\displaystyle f(x)=e^{−x^2}−1$$

$$\displaystyle \sum_{n=1}^∞\frac{(−1)^n}{n!}x^{2n}$$

2. $$\displaystyle f(x)=cosx−xsinx$$

## Exercise $$\PageIndex{9}$$

In the following exercises, find the Maclaurin series for $$\displaystyle F(x)=∫^x_0f(t)dt$$ by integrating the Maclaurin series of $$\displaystyle f(x)$$ term by term.

1. $$\displaystyle f(x)=\frac{sinx}{x}$$

$$\displaystyle F(x)=\sum_{n=0}^∞\frac{(−1)^n}{(2n+1)(2n+1)!}x^{2n+1}$$

2. $$\displaystyle f(x)=1−e^x$$

3. Use power series to prove Euler’s formula: $$\displaystyle e^{ix}=cosx+isinx$$

## Exercise $$\PageIndex{10}$$

The following exercises consider problems of annuity payments.

1. For annuities with a present value of $$\displaystyle 1$$ million, calculate the annual payouts given over $$\displaystyle 25$$ years assuming interest rates of $$\displaystyle 1%,5%$$, and $$\displaystyle 10%.$$

2. A lottery winner has an annuity that has a present value of $$\displaystyle 10$$ million. What interest rate would they need to live on perpetual annual payments of $$\displaystyle 250,000$$?

$$\displaystyle 2.5%$$
3. Calculate the necessary present value of an annuity in order to support annual payouts of $$\displaystyle 15,000$$ given over $$\displaystyle 25$$ years assuming interest rates of $$\displaystyle 1%,5%$$,and $$\displaystyle 10%.$$