I was asked to compute this series: ∑n=1∞1(2n−1)2 but by using the fact that ∑n=1∞1n2=π26

Nicontio1

Nicontio1

Answered

2021-12-23

I was asked to compute this series:
n=11(2n1)2
but by using the fact that n=11n2=π26

Answer & Explanation

scomparve5j

scomparve5j

Expert

2021-12-24Added 38 answers

By direct calculation, we see that
n=11n2=k=11(2k1)2+14n=11n2
which means
k=11(2k1)2=34n=11n2=34π26=π28
kalfswors0m

kalfswors0m

Expert

2021-12-25Added 24 answers

Observe that
n=11(2n)2=1(21)2+1(22)2+1(23)2=14(112+122+132+)=π24(6)
But
n=11(2n)2+n=11(2n1)2=n=11n2
where rearrangement is allowed since we are dealing with non-negative series.
user_27qwe

user_27qwe

Expert

2021-12-30Added 230 answers

Very detailed answer. Thanks.

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