Converges, Diverges, Can't Tell

• Nov 13th 2008, 07:39 AM
kl.twilleger
Converges, Diverges, Can't Tell
Answer with converges, diverges, or can't tell

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

So, since the $\displaystyle a_k$ 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?
• Nov 13th 2008, 08:39 AM
ThePerfectHacker
Quote:

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

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

So, since the $\displaystyle a_k$ 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 $\displaystyle a_n = \tfrac{1}{n}$. Then $\displaystyle \lim a_n = 0$ yet $\displaystyle \Sigma_{n\geq 1}a_n = \infty$.
• Nov 13th 2008, 10:38 AM
kl.twilleger
So then it diverges to infinity? Or do I not know?
• Nov 13th 2008, 10:51 AM
Plato
Quote:

Originally Posted by kl.twilleger
So then it diverges to infinity? Or do I not know?

The series may or may not converge.
Examples:
$\displaystyle \left( {\frac{1}{n}} \right) \to 0\;\& \;\sum\limits_{n = 1}^\infty {\frac{1}{n}}\mbox{ diverges.}$

$\displaystyle \left( {\frac{1}{n^2}} \right) \to 0\;\& \;\sum\limits_{n = 1}^\infty {\frac{1}{n^2}}\mbox{ converges.}$
• Nov 13th 2008, 12:43 PM
Mathstud28
Quote:

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

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

So, since the $\displaystyle a_k$ 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 $\displaystyle \sum{a_n}$ for which $\displaystyle a_{n}$ does not converge to zero then your series diverges. If $\displaystyle a_n$ 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 $\displaystyle a_n$ tends to zero but some series converge some don't[/tex]