I have \sum_{n=0}^\infty(-1)^n2nx^{(2n-1)} It turns out that this series is equal to

horrupavinvo

horrupavinvo

Answered question

2022-02-15

I have
n=0(1)n2nx(2n1)
It turns out that this series is equal to the function 2x(1+x2)2
Is there a general method that would demonstrate this fact beforehand?

Answer & Explanation

chung001vba

chung001vba

Beginner2022-02-16Added 6 answers

There is no single method for summing all power series. However, there is the sum of a geometric series:
k=0xk=11x
Substitution yields
k=0(1)kx2k=11+x2
Taking the derivative gives
k=0(1)k2kx2k1=2x(1+x2)2

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