I want to express the following power series 1 + x + x 2 </msup> + 3

grenivkah3z 2022-07-09 Answered
I want to express the following power series
1 + x + x 2 + 3 x 3 + x 4 + 5 x 5 + x 6 + 7 x 7 + x 8 + . .
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Answers (1)

postojahob
Answered 2022-07-10 Author has 13 answers
Your series is the sum of
1 + x 2 + x 4 + x 6 + = 1 1 x 2
with
x + 3 x 3 + 5 x 5 + = x ( 1 + 3 x 2 + 5 x 4 + ) = x ( x + x 3 + x 5 + ) = x ( x 1 x 2 ) = x 3 + x ( x 2 1 ) 2 ,
and therefore the sum of your series is
1 1 x 2 + x 3 + x ( x 2 1 ) 2 = x 3 x 2 + x + 1 ( x 2 1 ) 2 .
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