Question: "Study the behaviour of∑n=1∞n+log⁡(n)(n+cos⁡(n))3"

Nyah Conrad

Answered question

2022-02-27

Study the behaviour of

\sum_{n = 1}^{\infty} \frac{n + \log (n)}{{(n + \cos (n))}^{3}}

vazen2bl

Beginner2022-02-28Added 9 answers

For

n \geq 2

we have

\frac{n + \log (n)}{{(n + \cos (n))}^{3}} < \frac{n + n}{{(n - \frac{n}{2})}^{3}} = \frac{2 n}{\frac{n^{3}}{8}} = \frac{16}{n^{2}}

Thus

\sum_{n = 1}^{\infty} \frac{n + \log (n)}{{(n + \cos (n))}^{3}} < \frac{1}{1 + \cos (1)} + \sum_{n = 2}^{\infty} \frac{16}{n^{2}} = \frac{1}{1 + \cos (1)} + \frac{8 π^{2}}{3} - 16

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?