a problem on convergence of a series [duplicate]

Question

Possible Duplicate:
Consider the series $ ∑_{n=1}^∞ x^2+ n/n^2$ . Pick out the true statements:

Consider the series $ \sum_{n=1}^\infty (-1)^n(x^2+ n)/n^2$
Pick out the true statements:
(a) The series converges for all real values of $x$.
(b) The series converges uniformly on $\mathbb{R}$.

how can I able to solve this.

Answer 1

Splitting up the infinite series:

$$\sum_{n=1}^{\infty} (-1)^{n} (x^2 + n)/n^{2} = \sum_{n=1}^{\infty} (-1)^{n} x^{2}/n^{2} + \sum_{n=1}^{\infty} (-1)^{n} n/n^{2} = x^2 \sum_{n=1}^{\infty} (-1)^{n}/n^{2} + \sum_{n=1}^{\infty} (-1)^{n}/n$$

Note that each of the last two infinite sums converges (why?) which justifies the above line read from right to left. In particular, the sum of two convergent series converges.

Answer 2

You can apply alternating series test

http://en.wikipedia.org/wiki/Alternating_series_test