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Chapter 1: Q12P (page 36)

Show as in Problem 11that the Maclaurin series forexconverges toex.

Short Answer

Expert verified

The Maclaurin series for ex converges to ex.

Step by step solution

01

Given Information

The function is ex.

02

Definition of the convergence of the Series

The Series is said to be convergent if the terms of the Series are moving toward zero.

03

Verify the statement

To prove the Maclaurin series for exconverges to ex, we need to show that

fX=limn鈫�鈭�TnX which is equivalent to showing that the remainder

RnX=fX-TnX鈫�0. The function is fX=ex.

Plugging this into the inequality, and taking the limit, we have:

limn鈫�鈭�RnX鈮�limn鈫�鈭�xn+1exn+1!=exlimn鈫�鈭�xn+1n+1!=0

Hence, limn鈫�鈭�xn+1exn+1!=0andTnX鈫�fX.and .

Thus, the Maclaurin series for exconverges to ex.

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Most popular questions from this chapter

Find the interval of convergence of each of the following power series; be sure to investigate the endpoints of the interval in each case鈭�n=1鈭�x2n2nn2

In the following problems, find the limit of the given sequence as

n鈫�鈭�

Write the Maclaurin series for 11+xin鈭� form using the binomial coefficient notation. Then find a formula for the binomial coefficients in terms ofn as we did in Example 2 above

Use the special comparison test to find whether the following series converge or diverge.

鈭�n=9鈭�(2n+1)(3n-5)n2-73

Find the sum of each of the following series by recognizing it as the Maclaurin series for a function evaluated at a point.

鈭�n=0鈭�(-1)n(2n)!(蟺2)2n
See all solutions

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