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?

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 $\left[- 1 , 1\right]$, 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 $\frac{1}{n} ^ 2$ 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:

${\lim}_{n \to \infty} \frac{7 - \sin \left({n}^{2}\right)}{n} ^ 2 + 1 = 0 + 1 = 1$

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