How do you use the convergence tests, determine whether the given series converges #sum (7-sin(n^2))/n^2+1# from n to infinity?

2 Answers
May 23, 2015

If the general term is with the #1#, the series is divergent, if the #1# is not in the term, then follow me:

The function sinus has a range #[-1,1]#, so it doesn't influence the character of the series.

So:

#(7-sin(n^2))/n^2+1~1/n^2# (#~# means asyntotic).

Since the function #1/n^2# is convergent, so it is the given one.

May 23, 2015

First do the limit test for convergence, and check the limit, as the limit needs to equal 0 for a series to be considered convergent, if the limit is not 0, it is then divergent.

Thus:

#lim_(n -> oo) (7 - sin(n^2))/n^2 + 1 = 0 + 1 = 1#

Therefore we can conclude that the series does not converge, and therefore is Divergent.