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Calculus / Calculus: Early Transcendentals 1 / Chapter 8.3 / Problem 62E

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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Value of a seriesa. Find the value of the series

Chapter 11, Problem 62E

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QUESTION:

Values of a series

a. Find the value for the series

$\sum_{k=1}^{\infty} \frac{3^{k}}{\left(3^{k+1}-1\right)\left(3^{k}-1\right)}$

b. For what value of a does the series

$\sum_{k=1}^{\infty} \frac{a^{k}}{\left(a^{k+1}-1\right)\left(a^{k}-1\right)}$

Coverage, and in those cases, what is its value?

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QUESTION:

Values of a series

a. Find the value for the series

$\sum_{k=1}^{\infty} \frac{3^{k}}{\left(3^{k+1}-1\right)\left(3^{k}-1\right)}$

b. For what value of a does the series

$\sum_{k=1}^{\infty} \frac{a^{k}}{\left(a^{k+1}-1\right)\left(a^{k}-1\right)}$

Coverage, and in those cases, what is its value?

ANSWER:

SOLUTION

Step 1

We have to find the value of the series $\sum\limits_{k=1}^{\infty{}}\frac{{3}^{k}}{({3}^{k+1}-1)({3}^{k}-1)}$

Let us just substitute ${3}^{k}=x$ and do the partial decomposition.

Therefore $\frac{a}{(3a-1)(a-1)}=\frac{A}{3a-1}+\frac{B}{a-1}$

$\Rightarrow{}a=A(a-1)+B(3a-1)=a(A+3B)-(A+B)$

$A+B=0\ \ \ and\ \ \ \ A+3B=1$

If $\ a=1\ \ \Rightarrow{}B=\frac{1}{2}$

And therefore $A=-\frac{1}{2}$

Therefore we get $\frac{a}{(3a-1)(a-1)}=\frac{-1}{2(3a-1)}+\frac{1}{2(a-1)}=\frac{1}{2}(\frac{1}{a-1}-\frac{1}{3a-1})$

re substitute $a={3}^{k}$

Therefore $\sum\limits_{k=1}^{\infty{}}\frac{{3}^{k}}{({3}^{k+1}-1)({3}^{k}-1)}=\frac{1}{2}\sum\limits_{k=1}^{\infty{}}(\frac{1}{{3}^{k}-1}-\frac{1}{{3}^{k+1}-1})$

$\Rightarrow{}\ \ \ \ \frac{1}{2}((\frac{1}{2}+\frac{1}{8}+\frac{1}{26}+....)-(\frac{1}{8}+\frac{1}{26}+....))$

$\Rightarrow{}\frac{1}{2}\times{}\frac{1}{2}=\frac{1}{4}$

Therefore the value of $\sum\limits_{k=1}^{\infty{}}\frac{{3}^{k}}{({3}^{k+1}-1)({3}^{k}-1)}=\frac{1}{4}$

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Value of a seriesa. Find the value of the series

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Solution Found!

Value of a seriesa. Find the value of the series

Questions & Answers

Study Tools You Might Need

Chapter: 5 Problem: 82P

Chapter: 3 Problem: 38

Chapter: 4 Problem: 22E

Chapter: Problem: 12.53

Chapter: 31 Problem: 1

Chapter: 3 Problem: 2

Chapter: 6 Problem: 3

Chapter: 20 Problem: 3

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