0
$\begingroup$

Given:

$$ \sum_{n=4}^\infty \left(\frac {n^{2.5}-1}{n^{6.5}+4}\right) $$

I know that a given series converges if $ lim: a_n \to 0$, in the above case for ${n \to \infty}$ the limit is 0. Hence, it should converge. Am I right?

$\endgroup$
1
$\begingroup$

No, $\lim a_n=0$ is not sufficient for a series to converge. However, for large $n$ the series above is like $\sum 1/n^4$ and it converges because $\sum 1/n^A$ converges for all $A>1$.

$\endgroup$
  • $\begingroup$ It reduces to P series right? $\endgroup$ – vivek Mar 6 '16 at 17:13
  • $\begingroup$ The series can be compared to $\sum 1/n^4$ in the comparison test, I think. I'm not sure what a P series is. $\endgroup$ – ForgotALot Mar 6 '16 at 17:24
0
$\begingroup$

It is not true that a series converges if $\lim a_n=0$. For example, consider $$\sum_{n=1}^\infty\frac 1n.$$

Clearly, $\lim_{n\to\infty}\frac 1n=0$, but it is easily shown by the integral test that the series diverges.

$\endgroup$
  • $\begingroup$ Correct, it says that "If the series converges then lim is 0" its not a two way relationship. $\endgroup$ – vivek Feb 26 '16 at 2:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.