# Thread: Summability and convergence of partial sums

1. ## Summability and convergence of partial sums

I'm stuck on a problem here.
I have the sum of:
$\displaystyle ((-1)^t)/(t+1)$

How do I show that this is not summable? And what can we say about the convergence of its partial sums?

2. I have the sum of:
Do you mean $\displaystyle \sum_{t=0}^\infty\frac{(-1)^t}{t+1}$? This series converges because it is alternating and 1/(t+1) monotonically converges to 0 (Alternating series test). It does not converge absolutely, though, because taking the absolute value of each term makes the harmonic series.

3. Originally Posted by emakarov
Do you mean $\displaystyle \sum_{t=0}^\infty\frac{(-1)^t}{t+1}$? This series converges because it is alternating and 1/(t+1) monotonically converges to 0 (Alternating series test). It does not converge absolutely, though, because taking the absolute value of each term makes the harmonic series.
Yes. that's what I mean. Haven't learned the keyboard commands yet.
Well, I'm mainly concerned with the summability. I know the series converges, but the sum is unbounded, and thus cannot be dominated by another summable series for all t.
But I mainly want to know the best way to show a partial sum sequence that converges.

Thanks.

4. Originally Posted by emakarov
Do you mean $\displaystyle \sum_{t=0}^\infty\frac{(-1)^t}{t+1}$? This series converges because it is alternating and 1/(t+1) monotonically converges to 0 (Alternating series test). It does not converge absolutely, though, because taking the absolute value of each term makes the harmonic series.
Just got everything I needed from that Alternating Series Test link though. Thanks a bunch.