Express the series \[\sum_{n=1}^{\infty} \left(\frac{1}{(x-5)^{2n-1}} + \frac{1}{(x-4)^{2n-1}}\right)\]as a rational function.

Name: Express the series \[\sum_{n=1}^{\infty} \left(\frac{1}{(x-5)^{2n-1}}
Uploaded: 2024-03-20T07:06:16.000Z
Channel: Bartleby
Description: Express the series \[\sum_{n=1}^{\infty} \left(\frac{1}{(x-5)^{2n-1}} + \frac{1}{(x-4)^{2n-1}}\right)\]as a rational function.

Given: The series ∑n=1∞1x-52n-1+1x-42n-1, To find: The rational function expression for the given series.Explanation: It is known that the infinite sum of geometric series ∑n=0∞arn=a+ar+ar2+ar3+... =a1-r for r<1. Here the given series can be written as ∑n=1∞1x-52n-1+1x-42n-1=∑n=1∞1x-52n-1+∑n=1∞1x-42n-1=1x-5+1x-53+1x-55+...+1x-41+1x-43+1x-45+...=1x-51+1x-52+1x-54+...+1x-41+1x-42+1x-44+... hence here we have two geometric series 1+1x-52+1x-54+... with first term 1, common ratio 1x-52 and 1+1x-42+1x-44+... with first term 1, common ratio 1x-42. Finding the sum of these series, 1+1x-52+1x-54+...=11-1x-52=x-52x-52-1 for 1x-52<1⇒1x-52<1 ...(i) and 1+1x-42+1x-44+...=11-1x-42=x-42x-42-1 for 1x-42<1⇒1x-42<1 ...(ii) Hence from (i) and (ii) we have ∑n=1∞1x-52n-1+1x-42n-1=1x-51+1x-52+1x-54+...+1x-41+1x-42+1x-44+...=1x-5x-52x-52-1+1x-4x-42x-42-1 for 1x-52<1 and 1x-42<1 Simplifying the function =1x-5x-52x-52-1+1x-4x-42x-42-1, 1x-5x-52x-52-1+1x-4x-42x-42-1=x-5x-52-12+x-4x-42-12=x-5x-4x-6+x-4x-3x-5 ∵a2-b2=a+ba-b=x-52x-3+x-42x-6x-3x-4x-5x-6 Answer: ∑n=1∞1x-52n-1+1x-42n-1=x-52x-3+x-42x-6x-3x-4x-5x-6 for 1x-52<1 and 1x-42<1.