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Chapter 1: Q20P (page 41)

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

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

Short Answer

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

Step by step solution

01

Given Information

The given series are, i.e.,

(a)鈭�n=1鈭�n(4n2-1)2(b)鈭�n=1鈭�n3n!(c)鈭�n=1鈭�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 鈭�n=1鈭�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 鈭�n=1鈭�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 鈭�n=1鈭�n(n+1)3n,

Hence, the sum of the series 鈭�n=1鈭�n(n+1)3n=94.

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Most popular questions from this chapter

In the following problems, find the limit of the given sequence asn鈫�鈭�

10nn!

arcsinx=鈭�0xdt1-t2

1+ln(1+x)

Show that if p is a positive integer, thenpn=0when n>p , so (1+x)p=鈭�pnxnis just a sum ofp+1terms, from n=0 to n=p . For example,(1+x)2has 3 terms,(1+x)3 has4terms, etc. This is just the familiar binomial theorem.

f(x)=cos鈥�x,a=蟺.
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