Telescoping Series

Question

Decide whether each of the following series converges. If a given series converges, compute its sum. Otherwise, enter if it diverges to infinity, if it diverges to minus infinity, or diverges otherwise.

∞

∑=(sin(−6n)−sin(−6(n+1)))

n=1

 

 

Answer 1

The expansion gives you

sin(-6) - sin (-12) +

sin(-12) - sin(-18) +

sin(-18) - sin(-24 +

...

sin(-6n) - sin(-6n-6) etc.

 

All of the middle terms cancel (top right cancelled by succeeding term at left, etc...)

and at any instant (pausing with a particular n) we have sin (-6) - sin(-6n-6)

which at worst is bounded by -2 and by 2, so there's divergence of the OSCILLATION kind.

 

No particular sum can be offered.