Using the Ratio Test to see if a series converges or diverges?

**1. The problem statement, all variables and given/known data**

Use the Ratio Test for series to determine whether each of the following series converge or diverge. Make Reasoning Clear.

(a)

(b)

**2. Relevant equations**

**3. The attempt at a solution**

(a) I let = sequence of partial sums then plugged everything into ratio test formula.

I ended up with:

I know that the limit equals 0 hence the series converges but not quite sure how to show that the limit cancels down to show 0 is "obvious"

Any help would be great, thanks.

(b) I let = sequence of partial sums then plugged everything into ratio test formula.

I ended up with:

I know that:

And that:

Hence the overall limit = infinity and so the series diverges; but, once again i'm not quite sure how to show that how the limits cancels down to show ∞ and 0 is "obvious"

Once again, any help would be great, thanks.

Re: Using the Ratio Test to see if a series converges or diverges?

For a) notice that and . Then

Re: Using the Ratio Test to see if a series converges or diverges?

Thanks, that's very good and I understand all of it; however, I haven't been formally taught L'Hospitals Rule yet and so am not allowed to use it for this particular problem. :(

Any other ideas?

Re: Using the Ratio Test to see if a series converges or diverges?

Say that it's self-taught then.

Re: Using the Ratio Test to see if a series converges or diverges?

Could I equally get away with saying that since the first part of the limit = 0 then it doesn't matter what the second part = since it will still be zero?

Re: Using the Ratio Test to see if a series converges or diverges?

Quote:

Originally Posted by

**lmstaples** Could I equally get away with saying that since the first part of the limit = 0 then it doesn't matter what the second part = since it will still be zero?

No, because is an entirely different kettle of fish.

I expect you would have to make use of somehow...

Re: Using the Ratio Test to see if a series converges or diverges?

Now that standard limit I have been taught and can use. I was trying to implement it before buy didn't get anywhere - I should have kept with it longer.

Re: Using the Ratio Test to see if a series converges or diverges?

Hello, Imstaples!

Another approach . . .

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Re: Using the Ratio Test to see if a series converges or diverges?

To the OP, I like very much the way you presented the problem. (Clapping) Let me offer a small tip for your use of :

i) Use the code \infty to get

ii) Use the code \lim_{n\to\infty} to get