Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
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=0anxnn=0anxn is 5,5, then the radius of convergence for the series n=1nanxn1n=1nanxn1 is also 5.5.

254.

Power series can be used to show that the derivative of exisex.exisex. (Hint: Recall that ex=n=01n!xn.)ex=n=01n!xn.)

255.

For small values of x,sinxx.x,sinxx.

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 $1$1 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 $10$10 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 $15,000$15,000 given over 2525 years assuming interest rates of 1%,5%,and10%.1%,5%,and10%.

Order a print copy

As an Amazon Associate we earn from qualifying purchases.

Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction
Citation information

© Feb 5, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.