Show whether the series ((-1)^n)*((2^n)/(n^4))) is absolutely convergent.
Taking an=((-1)^n)*((2^n)/(n^4))) , here |an|=((2^n)/(n^4)).
I have the gut feeling that this is absolutely convergent. But don't see a way to do that. Also, we can only use comparison test, monotone convegence theorem, convergent==>boundedness, convergent sequence has only one limit.
To show |an| is absolutely convergent(which will also prove an is convergent), I need to find another bn, which converges, such that an<=bn. I know bn=n!/n^4 will do, but need to prove that converges first. How can I prove this?
Maybe we can use comparison test seperately more than once right?
Any thoughts on solving this?
Thanks!
exponentials grow faster than powers.
For sufficiently large n, 2^n/n^4 >1
and things explode.
and n doesn't even have to be very big.