# Thread: Converges, Diverges, Can't Tell

1. ## Converges, Diverges, Can't Tell

Answer with converges, diverges, or can't tell

$If the \lim a_k=0, then \sum a_k=0$

So, since the $a_k$ isn't an absolute value, doesn't that mean that I can't tell if it diverges or converges? Or is there some other rule that I'm forgetting when it comes to determining converging/diverging?

2. Originally Posted by kl.twilleger
Answer with converges, diverges, or can't tell

$If the \lim a_k=0, then \sum a_k=0$

So, since the $a_k$ isn't an absolute value, doesn't that mean that I can't tell if it diverges or converges? Or is there some other rule that I'm forgetting when it comes to determining converging/diverging?
Let $a_n = \tfrac{1}{n}$. Then $\lim a_n = 0$ yet $\Sigma_{n\geq 1}a_n = \infty$.

3. So then it diverges to infinity? Or do I not know?

4. Originally Posted by kl.twilleger
So then it diverges to infinity? Or do I not know?
The series may or may not converge.
Examples:
$\left( {\frac{1}{n}} \right) \to 0\;\& \;\sum\limits_{n = 1}^\infty {\frac{1}{n}}\mbox{ diverges.}$

$\left( {\frac{1}{n^2}} \right) \to 0\;\& \;\sum\limits_{n = 1}^\infty {\frac{1}{n^2}}\mbox{ converges.}$

5. Originally Posted by kl.twilleger
Answer with converges, diverges, or can't tell

$If the \lim a_k=0, then \sum a_k=0$

So, since the $a_k$ isn't an absolute value, doesn't that mean that I can't tell if it diverges or converges? Or is there some other rule that I'm forgetting when it comes to determining converging/diverging?
Are you trying to remember the n-th term test for divergence, which roughly states that if you have $\sum{a_n}$ for which $a_{n}$ does not converge to zero then your series diverges. If $a_n$ does converge to zero this says nothing of the series convergence or divergence as can be shown by TPH or Plato's or a hundred million other cases where $a_n$ tends to zero but some series converge some don't[/tex]