How to prove the divergence of the following sum: <munderover> &#x2211;<!-- ∑ --> <mro

prensistath

prensistath

Answered question

2022-05-30

How to prove the divergence of the following sum: x = 1 cot 1 [ 1 1 + x 2 + x ]

Answer & Explanation

Ada Harrington

Ada Harrington

Beginner2022-05-31Added 12 answers

Recall that arccot 0 = π 2 , hence, for x
arccot 1 1 + x + x 2 arccot 0 = π 2 0
So, as the sequence of the terms does not converge to zero, the series cannot converge.
rs450nigglix2

rs450nigglix2

Beginner2022-06-01Added 3 answers

HINT:
arccot 1 1 + x + x 2 = π 2 arctan 1 1 + x + x 2
arctan 1 1 + x + x 2 = arctan x + 1 x 1 + x ( 1 + x ) = arctan ( x + 1 ) arctan x
which is a Telescoping Series

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