Question

Determine convergence or divergence for this series ? sin k/(k+1)!

Answer 1

The series is absolutely convergent.

Explanation:

We know that:

`abs (sinx) <=1` for any `x in RR`

Then:

`abs (sink)/((k+1)!) <= 1/((k+1)!)`

Now we can prove that the series:

`sum_(n=0)^oo 1/((k+1)!)`

is convergent using the ratio test, as:

`lim_(k->oo) (1/((k+2)!))/(1/((k+1)!)) = lim_(k->oo) ((k+1)!)/((k+2)!) = lim_(k->oo) 1/(k+2) = 0`

and then by direct comparison also the series:

`sum_(n=0)^oo abs (sink)/((k+1)!)`

is convergent,