Solved: "Does the series ∑n=1∞sin⁡(n−n2+n)n converge?"

afasiask7xg

Answered question

2022-04-02

Does the series

\sum_{n = 1}^{\infty} \frac{\sin (n - \sqrt{n^{2} + n})}{n}

converge?

Nunnaxf18

Beginner2022-04-03Added 18 answers

The key here is that

n - \sqrt{n^{2} + n}

converges to

- \frac{1}{2}

as n goes to infinity:

n - \sqrt{n^{2} + n} = (n - \sqrt{n^{2} + n}) \times \frac{n + \sqrt{n^{2} + n}}{n + \sqrt{n^{2} + n}}

= \frac{n^{2} - (n^{2} + n)}{n + \sqrt{n^{2} + n}} = - \frac{n}{n + \sqrt{n^{2} + n}}

= - \frac{1}{1 + \sqrt{1 + \frac{1}{n}}}

Take limits as n goes to infinity to get

- \frac{1}{2}

Hence

\sin (n - \sqrt{n^{2} + n})

converges to

\sin (- \frac{1}{2})

, and the series diverges similarly to

\frac{1}{n}

, using the limit comparison test for example.

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