Calculate the sum function for the power series <munderover> &#x2211;<!-- ∑ --> <mrow cla

trojanman849089tck

trojanman849089tck

Answered question

2022-06-04

Calculate the sum function for the power series n = 0 1 4 n 2 1 x 2 n + 1

Answer & Explanation

Siffkiffanewsler6n

Siffkiffanewsler6n

Beginner2022-06-05Added 7 answers

Recall the following Taylor expansions:
log ( 1 + x ) = k = 1 ( 1 ) k + 1 x k k log ( 1 x ) = k = 1 x k k
Adding them together you are left with two copies of each term with odd powers:
log ( 1 + x ) log ( 1 x ) = 2 ( x + x 3 3 + x 5 5 + ) = 2 k = 1 x 2 k 1 2 k 1 = 2 x 1 k = 1 x 2 k 2 k 1 = 2 ( f ( x ) + 1 ) x

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