For every k ∈ N , let x k = ∑ n = 1 ∞...

hornejada1c

hornejada1c

Answered

2022-07-11

For every k N , let
x k = n = 1 1 n 2 ( 1 1 2 n + 1 4 n 2 ) 2 k .

Answer & Explanation

escampetaq5

escampetaq5

Expert

2022-07-12Added 12 answers

Solution 1: By the dominated convergence theorem, we may switch the order of the limit and the sum, so we see that
lim k x k = 0.
Solution 2: Notice the terms are always bounded above by 1 n 2 . Let ϵ > 0, and choose N such that
n = N 1 n 2 < ϵ .
Then choose k so large that
( 1 1 2 N ) 2 k ϵ N .
It then follows that | x k | 2 ϵ , and the same inequality holds for all j k. Since ϵ was arbitrary the proof is finished.

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