How to prove that <munderover> &#x2211;<!-- ∑ --> <mrow class="MJX-TeXAtom-ORD"> n

Kendrick Hampton

Kendrick Hampton

Answered question

2022-06-22

How to prove that n = 0 N 2 2 ( 1 ) n ( N 2 n 1 2 n + 1 ) N π 4 for even N ?

Answer & Explanation

Jaylee Dodson

Jaylee Dodson

Beginner2022-06-23Added 22 answers

Assuming N = 2 M we have to prove:
(1) n = 0 M 1 ( 1 ) n ( 2 M 2 n + 1 1 ) = ( 1 ) M 1 2 + 2 M n = 0 M 1 ( 1 ) n 2 n + 1 M π 2 .
It is well known that:
(2) n = 0 + ( 1 ) n 2 n + 1 = π 4
hence if M is even we have:
(3) n = 0 M 1 ( 1 ) n 2 n + 1 < π 4
while for odd Ms we have:
(3) 1 2 M + n = 0 M 1 ( 1 ) n 2 n + 1 < π 4
hence (1) is proved.

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