Question

What is the convergence (absolute or conditional) of the alternate series: `f(x)=sum_{n=1} ^oo (-1)^n ((n)/(7n^2+3))` ?

Answer 1

This series is conditionally convergent but not absolutely convergent.

Explanation:

Since this series is alternating, it will conditionally converge provided the individual terms tend to `0`, which they do:

`lim_(n->oo) (n/(7n^2+3)) = lim_(n->oo)1/(7n+3/n) = 0`

On the other hand, it is not absolutely convergent, since when `n >= 1` we have:

`7n^2+3 <= 10n^2`

So:

`n/(7n^2+3) >= n/(10n^2) = 1/10 1/n`

So:

`sum_(n=1)^oo abs((-1)^n (n/(7n^2+3))) >= 1/10 sum_(n=1)^oo 1/n = oo`