I need to show that the series of general term tanh 1/n+ ln(n^2-n)/(n^2+1) converges.

Christopher Saunders

Christopher Saunders

Answered question

2022-11-01

Proving convergence of a series
tanh 1 n + ln n 2 n n 2 + 1

Answer & Explanation

rcampas4i

rcampas4i

Beginner2022-11-02Added 22 answers

You should keep the error term in your computations. In particular in the taylor approximation of ln you should keep the term 1 n 2 .
tanh 1 n + ln n 2 n n 2 + 1 = ( 1 n + o ( 1 n 2 ) ) + ( 1 n 3 2 n 2 + o ( 1 n 2 ) ) = 3 2 n 2 + o ( 1 n 2 ) .
In general a series of the form
n c n α + o ( 1 n α )
converges if α > 1 and diverges if α 1

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