Why the following series ∑ n = 1 ∞ ln ⁡ n n 3 − n + 1 converges?

Question

Why the following series      ∑          n      =      1              ∞                  ln      ⁡      n                      n        3            −      n      +      1      converges?

Savion Chaney · Accepted Answer

Note that by L&#039;Hospital&#039;s Rule, or other methods,       lim          n      →      ∞                  log      ⁡      n              n              1                  /                4              =  0. So if n is large enough,   log  ⁡  n  &amp;lt;      n          1              /            4      Note also that       n    3    −  n  +  1  &amp;gt;      n    3        /    4. Thus if n is large enough, then            log      ⁡      n                      n        3            −      n      +      1        &amp;lt;      2          n              5                  /                4              .It follows by Comparison that our series converges.

Howard Nelson · Accepted Answer

A related problem. Note that, you can compare as

\frac{\ln n}{\sqrt{n^{3} - n + 1}} \sim \frac{\ln n}{\sqrt{n^{3}}},

and the series

\sum_{n} \frac{\ln n}{\sqrt{n^{3}}}

converges by integral test.

Why the following series sum_(n=1)^infty ln n/(sqrt(n^3-n+1)) converges?

Answered question

Answer & Explanation

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