Does the series ∑[n=1,∞) sin4n / 4^n converge or diverge?

[Rossinole]

Homework Statement

Does the series ∑[n=1,∞) sin4n / 4^n converge or diverge?

[h2]Homework Equations[/h2]

Ratio Test

lim n->∞ | a_n+1 / a_n |

The Attempt at a Solution

By Ratio Test.

Let a_n = sin(4n) / 4^n

So,

lim n->∞ | (sin (4n+1) / 4^n+1) / (sin 4n / 4^n) |

Skipping a few steps..

= | (sin(4n+1)/sin(4n)) * (4^n)/(4^n * 4^1) |

= 1/4 * lim n->∞ (sin(4n+1)/sin(4n))

Here's my problem. How do I take the limit of (sin(4n+1)/sin(4n))? Did I do the whole problem wrong? Should I have used Root test?

 

[CompuChip]

I don't think the limit works out, as both sines will keep on oscillating, it's just like an ordinary sine in that respect. I think you will find the root test more useful, as you suggested.

 

[Rossinole]

Thank you, I will try that.