Expert Answer to "Prove that:∑k=1∞k!sin⁡(k)kkconvergence or divergence?"

Arslan Lyons

Answered question

2022-02-25

Prove that:

\sum_{k = 1}^{\infty} \frac{k! \sin (k)}{k^{k}}

convergence or divergence?

Daisie Benitez

Beginner2022-02-26Added 5 answers

The ratio test gives the value

\frac{1}{e}

for

lim \frac{a_{n + 1}}{a_{n}}

So the series

\sum_{n = 1}^{\infty} \frac{n! \cdot 1}{n^{n}}

is convergent. Seems you just miscalculated.
So the given series

\sum_{k = 1}^{\infty} \frac{k! \cdot \sin (k)}{k^{k}}

is absolutely convergent.

husudiwareh

Beginner2022-02-27Added 7 answers

Remark that :

∣ \frac{k! \sin k}{k^{k}} \leq \frac{k!}{k^{k}}

= \frac{1 \times 2 \times \dots \times k}{k \times k \times \dots \times k}

\leq \frac{2}{k^{2}}

and the series:

\sum \frac{2}{k^{2}}

converges.

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