How do I determine that this series sum_(n=1)^(oo) log(1+(x)/(n^3)) converges pointwise for −1<x<1?

Juan Leonard

Juan Leonard

Answered question

2022-10-28

How do I determine that this series n = 1 log ( 1 + x n 3 ) converges pointwise for 1 < x < 1?

Answer & Explanation

scranna0o

scranna0o

Beginner2022-10-29Added 16 answers

If x = 0, it converges.
if x 0, we use the equivalence
ln ( 1 + a ) a ( a 0 )
which gives
ln ( 1 + a ) a ( a 0 )
and since 1 n 3 converges, we conclude that x n 3 and ln ( 1 + x n 3 ) converge.

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