Prove this series converges

Hi,

I have the following question:

__Part d__

Using appropriate series convergence tests, prove that the series

converges if and only if the real number satisfies

So my current working is this:

Let , then

Now as tends to infinity, will tend to .

Therefore, by the ratio test:

If , the series converges

If , the series diverges

If ,

In part b of this same question, we had to prove that diverges, which I did successfully.

According to my working then, the series converges if , which only partly satisfies the requirement:

Can someone spot what I've done wrong here?

Thanks

Note: In part c of the same question, we had to state and prove the Alternating Series Test, which I did successfully, so I'm thinking that we're meant to apply it here somehow, but I'm not sure how.

Re: Prove this series converges

Actually what you want using the ratio test is

which gives or . Then check your endpoints giving the two series

and

Then determine the convergence of each one to determine whether these endpoints should be included in your interval.

Re: Prove this series converges

Quote:

Originally Posted by

**Danny** Actually what you want using the ratio test is

which gives

or

. Then check your endpoints giving the two series

and

Then determine the convergence of each one to determine whether these endpoints should be included in your interval.

Thanks.

According to Alternating series test - Wikipedia, the free encyclopedia, the Alternating Series Test is from , not . Does this make a difference?

I previously knew the Alternating Series Test as

Is this the same as what Wikipedia says?

Thanks again

Re: Prove this series converges

Re: Prove this series converges

Quote:

Originally Posted by

**Danny** You'll notice that

is not defined at

so it should start at

As for the test, it doesn't matter shere you start. Adding or subtracting a few terms from the series won't change whether the series converges or not.

Is "my version" of the alternating series test:

wrong then?

Re: Prove this series converges

Nope - it doesn't matter.