Do these Series Converge or Diverge?
Decide whether each of the following series converges. If a given series converges, compute its sum. Otherwise, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, and DIV otherwise.
1) sum from n=1 to INF (e^(10n)-e^(10(n+1))
2) sum from n=1 to INF (sin(14n)-sin(14(n+1)))
3) sum from n=1 to INF (sin(14/n) - sin (14/(n+1)))
They are all telescopic series so for the first one I've reasoned that it will be e^(10)-e^(10(n+1) as n is increasing and will go to MINF. Is this correct?
No ideas for the second one.
For the third one, I reason it is sin (14) - sin (14/(n+1)) as n goes to infinity, which will be 0 and sin (0) = 0 sor the final answer would be sin (14). Is this correct?
More Questions From This User See All
Answers & Comments
Verified answer
They're all telescoping.
1) Diverges because the partial sums always look like S_N=e^10-e^(10(N+1)) which diverges in limit (to negative infinity).
2) Diverges because the partial sums always look like S_N=sin(14)-sin(14(N+1)), and sin(14(N+1)) can't make its mind up about settling on a particular value.
3) Converges because the partial sums always look like S_N=sin(14)-sin(14/(N+1)), and sin(14/(N+1)) is sin(0) in limit, which is just 0, so the sum is sin(14).
diverge