Question

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?

Answer 1

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.

Answer 2

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.