I am trying to show that the series <munderover> &#x2211;<!-- ∑ --> <mrow class="MJX-TeX

Loreenwk

Loreenwk

Answered question

2022-05-22

I am trying to show that the series
n = 1 z n 1 ( 1 z n ) ( 1 z n + 1 )

Answer & Explanation

Louis Lawrence

Louis Lawrence

Beginner2022-05-23Added 10 answers

I think you made a typo. Partial sums a given by the formula
n = 1 m z n 1 ( 1 z n ) ( 1 z n + 1 ) = z m 1 ( z 1 ) 2 ( z m + 1 1 )
If | z | < 1, then lim m z m = 0, so
n = 1 z n 1 ( 1 z n ) ( 1 z n + 1 ) = lim m n = 1 m z n 1 ( 1 z n ) ( 1 z n + 1 ) = lim m z m 1 ( z 1 ) 2 ( z m + 1 1 ) =
0 1 ( z 1 ) 2 ( 0 z 1 ) = 1 ( z 1 ) 2
If | z | > 1, then lim m z m = 0, so
n = 1 z n 1 ( 1 z n ) ( 1 z n + 1 ) = lim m n = 1 m z n 1 ( 1 z n ) ( 1 z n + 1 ) = lim m z m 1 ( z 1 ) 2 ( z m + 1 1 ) =
lim m 1 z m ( z 1 ) 2 ( z z m ) = 1 0 ( z 1 ) 2 ( z 0 ) = 1 z ( z 1 ) 2

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