Cauchy condensation test - do I have to prove <munderover> &#x2211;<!-- ∑ --> <mrow class

seiyakou2005n1

seiyakou2005n1

Answered question

2022-05-25

Cauchy condensation test - do I have to prove n = 2 1 n 2 converges?

Answer & Explanation

Sasha Pacheco

Sasha Pacheco

Beginner2022-05-26Added 10 answers

I do need to prove converges of:
n = 2 1 n 2
I am doing so by using once more Cauchy condensation test: I will call
a k = 1 k 2
Thus, by using Cauchy condensation test( n = 0 2 k a 2 k ), we get:
a 2 k = 1 n 2
Then use the formula of Cauchy condensation test itself, and we get:
n = 2 1 2 k 2 k 2 k
Which equals to:
n = 2 1 2 k
And here we see it. The n = 2 1 2 k converges.

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