Series Convergence

[cheffy]

If the limit of a series is a finite number, does that automatically mean that it converges?

For example, I have \(\displaystyle \[
a_n = \frac{{1 - 5n^4 }}{{n{}^4 + 8n^3 }}
\]\)

I calculated its limit from n to infinity to be -5. Is that all I have to do to prove that it converges? Or is there something else?
Thanks!

 

[stapel]

It sounds as though you may be confusing two different things: the limit of a series, and the limit of the terms in the sequence that forms the series.

In your example, the terms, a_n, do indeed have a limit value of -5. But does -5 - 5 - 5 - 5 - ... - 5 - 5 - 5 - ... converge to any finite value?

Remember: The sequence is just the terms; the series is the sum of all of those terms.

Can a series have a finite value if its sequence's terms have a non-zero limit (and all the terms have the same sign)?

Eliz.

 

[cheffy]

oops!

You're right. I meant to say sequence instead of series. I'm looking to find if the sequence converges or diverges.

 

[pka]

Yes the sequence has a limit of -5.