Calculus Volume 2

# Review Exercises

Calculus Volume 2Review Exercises

### 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 .

$∑ n = 0 ∞ n 2 ( x − 1 ) n ∑ n = 0 ∞ n 2 ( x − 1 ) n$

258 .

$∑ n = 0 ∞ x n n n ∑ n = 0 ∞ x n n n$

259 .

$∑ n = 0 ∞ 3 n x n 12 n ∑ n = 0 ∞ 3 n x n 12 n$

260 .

$∑ n = 0 ∞ 2 n e n ( x − e ) n ∑ n = 0 ∞ 2 n e n ( x − e ) n$

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 .

$f ( x ) = x 2 x + 3 f ( x ) = x 2 x + 3$

262 .

$f ( x ) = 8 x + 2 2 x 2 − 3 x + 1 f ( x ) = 8 x + 2 2 x 2 − 3 x + 1$

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

263 .

$f ( x ) = tan −1 ( 2 x ) f ( x ) = tan −1 ( 2 x )$

264 .

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

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 .

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

266 .

$f ( x ) = e 1 / ( 4 x ) , a = 4 f ( x ) = e 1 / ( 4 x ) , a = 4$

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

267 .

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

268 .

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

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

269 .

$f ( x ) = sin x , a = π 2 f ( x ) = sin x , a = π 2$

270 .

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

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

271 .

$f ( x ) = e − x 2 − 1 f ( x ) = e − x 2 − 1$

272 .

$f ( x ) = cos x − x sin x f ( x ) = cos x − x sin x$

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 .

$f ( x ) = sin x x f ( x ) = sin x x$

274 .

$f ( x ) = 1 − e x f ( x ) = 1 − e x$

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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