Does the series <munderover> &#x2211;<!-- ∑ --> <mrow class="MJX-TeXAtom-ORD"> n

lobht98

lobht98

Answered question

2022-06-26

Does the series n = 1 log ( n + 1 n ) converge?

Answer & Explanation

Govorei9b

Govorei9b

Beginner2022-06-27Added 21 answers

Your argument that b n 0 can be made much simpler: n + 1 n 1 and since log is continuous so log ( n + 1 n ) log 1 = 0. However this won't help as your proof breaks down after that: you say that there is some convergent a n such that a n L 0 but this is not possible as if a n converges then a n 0
We know that log x x 1 for x 1 (as follows from the Taylor series expansion of log x). Since k + 1 k 1 this means that your series b n behaves roughly like ( k + 1 k 1 ) = 1 k and hence diverges. To prove this rigorously use the comparison test with n = 1 1 n
Another way to prove divergence is by writing the partial sums of the series and noticing that by telescoping all elements cancel and you are left with k = 1 n log ( k + 1 k ) = log ( n + 1 )

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