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Chapter 1: Q20P (page 41)

By computer or tables, find the exact sum of each of the following series.

(a)∑n=1∞n(4n2-1)2(b)∑n=1∞n3n!(c)∑n=1∞n(n+1)3n

Short Answer

Expert verified
  1. The sum of the series∑n=1∞n(4n2-1)2=18.
  2. The sum of the series∑n=1∞n3n!=13.5914091422=5e.
  3. The sum of the series ∑n=1∞n(n+1)3n=94.

Step by step solution

01

Given Information

The given series are, i.e.,

(a)∑n=1∞n(4n2-1)2(b)∑n=1∞n3n!(c)∑n=1∞n(n+1)3n

02

Definition of Sum of a Series

The total of all the terms in a sequence can be solved as the sum of a series.

03

(a) Find the exact sum of the series ∑n=1∞n(4n2-1)2

Use the MATLAB tool to calculate the exact sum of the series,

Hence, the sum of the series ∑n=1∞n(4n2-1)2=18.

04

(b) Find the exact sum of the series ∑n=1∞n3n!

Use the MATLAB tool to calculate the exact sum of the series,

Hence, the sum of the series ∑n=1∞n3n!=13.5914091422=5e.

05

(c) Find the exact sum of the series ∑n=1∞n(n+1)3n

Use the MATLAB tool to calculate the exact sum of the series ∑n=1∞n(n+1)3n,

Hence, the sum of the series ∑n=1∞n(n+1)3n=94.

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Most popular questions from this chapter

Prove theorem 14.3. Hint: Group the terms in the error as role="math" localid="1657423688910" (an+1+an+2)+(an+3+an+4)+⋯to show that the error has the same sign as role="math" localid="1657423950271" an+1.Then group them asrole="math" localid="1657423791335" an+1+(an+2+an+3)+(an+4+an+5)+⋯to show that the error has magnitude less than|an+1|.

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

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

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

d4dx4ln(1+x3)atx=0 .

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

limx→0In(1-x)x .
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