1

I understand that given the absolute convergence test, if I am able to prove that the absolute of the series converges, then the series itself will converge itself as well.

What if I want to prove for conditional convergence? Is it sufficient to prove straight away that the absolute does not converge, or do I have to first prove that the series converges, and then prove that absolute does not converge, hence it must converge conditionally?

statsguy21
  • 432

2 Answers2

1

Yes, of course you need to prove both. If you prove the series of absolute values does not converge it doesn't mean that the original series converge. For example the series $\sum_{n=0}^\infty (-1)^n$ simply diverges.

Mark
  • 39,605
1

A series converges conditionally if both of the following conditions are true:

  1. The series converges
  2. The series does not converge absolutely.

To prove a series converges absolutely, you need to prove both conditions. It is not enough to prove that the series does not converge absolutely, since, for example, $$\sum_{n=1}^\infty (-1)^n n$$ does not converge absolutely, but that doesn't mean it converges conditionally.

5xum
  • 123,496
  • 6
  • 128
  • 204
  • I have an example on hand. 1 + sin$(\frac{1}{2})$ + sin$(\pi + \frac{1}{2}) + ...$. Since sin$(\frac{1}{n})$ is simply the negative of sin$(\pi + \frac{1}{n})$, I see that the series converges to 1. In this case, am I right to say that this series converges conditionally but not absolutely? – statsguy21 Oct 09 '18 at 09:11
  • @statsguy21 Did you prove that the series doesn't converge absolutely? So far, you only proved (well, sort of proved, there's still some work to do) that the series converges. – 5xum Oct 09 '18 at 09:15