Starting with the geometric series \sum_{n=0}^\infty x^n, find the sum of the series\sum_{n=1}^\infty nx^{n-1},\ |x|<1

Harlen Pritchard

Harlen Pritchard

Answered question

2021-06-13

Starting with the geometric series n=0xn, find the sum of the series
n=1nxn1, |x|<1

Answer & Explanation

Anonym

Anonym

Skilled2021-06-14Added 108 answers

Consider the geometric series,
n=0xn=1+x+x2+x3+...
=11x
Find the sum of the series n=1nxn1, |x|<1
n=1nxn1=1+2x+3x2+4x3+...
=(1+x+x2+x3+...)+(x+2x2+3x2+4x3+...)
=11x+x(1+2x+3x2+4x3+...)
=11x+xn=1nxn1
n=1nxn1xn=1nxn1=11x
(1x)n=1nxn1=11x
n=1nxn1=1(1x)2

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