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Calculus Volume 2

# Chapter Review Exercises

Calculus Volume 2Chapter Review Exercises

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

253.

If the radius of convergence for a power series $∑n=0∞anxn∑n=0∞anxn$ is $5,5,$ then the radius of convergence for the series $∑n=1∞nanxn−1∑n=1∞nanxn−1$ is also $5.5.$

254.

Power series can be used to show that the derivative of $exisex.exisex.$ (Hint: Recall that $ex=∑n=0∞1n!xn.)ex=∑n=0∞1n!xn.)$

255.

For small values of $x,sinx≈x.x,sinx≈x.$

256.

The radius of convergence for the Maclaurin series of $f(x)=3xf(x)=3x$ is $3.3.$

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

257.

$∑n=0∞n2(x−1)n∑n=0∞n2(x−1)n$

258.

$∑n=0∞xnnn∑n=0∞xnnn$

259.

$∑n=0∞3nxn12n∑n=0∞3nxn12n$

260.

$∑n=0∞2nen(x−e)n∑n=0∞2nen(x−e)n$

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.

261.

$f(x)=x2x+3f(x)=x2x+3$

262.

$f(x)=8x+22x2−3x+1f(x)=8x+22x2−3x+1$

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

263.

$f(x)=tan−1(2x)f(x)=tan−1(2x)$

264.

$f(x)=x(2+x2)2f(x)=x(2+x2)2$

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?

265.

$f(x)=x3−2x2+4,a=−3f(x)=x3−2x2+4,a=−3$

266.

$f(x)=e1/(4x),a=4f(x)=e1/(4x),a=4$

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

267.

$f(x)=cos(3x)f(x)=cos(3x)$

268.

$f(x)=ln(x+1)f(x)=ln(x+1)$

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

269.

$f(x)=sinx,a=π2f(x)=sinx,a=π2$

270.

$f(x)=3x,a=1f(x)=3x,a=1$

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

271.

$f(x)=e−x2−1f(x)=e−x2−1$

272.

$f(x)=cosx−xsinxf(x)=cosx−xsinx$

In the following exercises, find the Maclaurin series for $F(x)=∫0xf(t)dtF(x)=∫0xf(t)dt$ by integrating the Maclaurin series of $f(x)f(x)$ term by term.

273.

$f(x)=sinxxf(x)=sinxx$

274.

$f(x)=1−exf(x)=1−ex$

275.

Use power series to prove Euler’s formula: $eix=cosx+isinxeix=cosx+isinx$

The following exercises consider problems of annuity payments.

276.

For annuities with a present value of $11$ million, calculate the annual payouts given over $2525$ years assuming interest rates of $1%,5%,and10%.1%,5%,and10%.$

277.

A lottery winner has an annuity that has a present value of $1010$ million. What interest rate would they need to live on perpetual annual payments of $250,000?250,000?$

278.

Calculate the necessary present value of an annuity in order to support annual payouts of $15,00015,000$ given over $2525$ years assuming interest rates of $1%,5%,and10%.1%,5%,and10%.$

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