You are Here: Home >< Maths

# Does this need Abel's theorem? watch

1. I wanted to link to this thread earlier, but I didn't want to ruin the hint!
2. (Original post by Piecewise)
I wanted to link to this thread earlier, but I didn't want to ruin the hint!
Although it still doesn't show differentiating the sum does what you want, and to be honest I'm not convinced the OP has either. (Weierstrauss M-test shows the result of differentiating term by term converges uniformly, but that doesn't directly prove it converges to the result of differentiating the expression for the sum).

Edit: (to be clear, what you've posted is fine for normal pre-uni maths, but it isn't a proof).
3. (Original post by DFranklin)
Edit: (to be clear, what you've posted is fine for normal pre-uni maths, but it isn't a proof).
Just to be safe, is it fine, say, for the STEP?
4. (Original post by DFranklin)
Although it still doesn't show differentiating the sum does what you want, and to be honest I'm not convinced the OP has either. (Weierstrauss M-test shows the result of differentiating term by term converges uniformly, but that doesn't directly prove it converges to the result of differentiating the expression for the sum).
It's halfway there though. I remember a first-year proof without uniform convergence. It was hard and ugly. Then in second year we got a much easier proof, admittedly assuming the fundamental theorem of calculus among other things.
5. (Original post by Piecewise)
Just to be safe, is it fine, say, for the STEP?
Yes.
6. (Original post by Zhen Lin)
It's halfway there though. I remember a first-year proof without uniform convergence. It was hard and ugly.
It's really not that hard to find the finite sum and then take the limit.

Then in second year we got a much easier proof, admittedly assuming the fundamental theorem of calculus among other things.
Sure. But as I said, it's not exactly a "obvious" consequence of uniform convergence.
7. (Original post by DFranklin)
It's really not that hard to find the finite sum and then take the limit.
Hmmm. It wasn't that easy, as I recall... Lemma 9.6 and theorem 9.7 here (p.40-41).
8. (Original post by Zhen Lin)
Hmmm. It wasn't that easy, as I recall... Lemma 9.6 and theorem 9.7 here (p.40-41).
Well, it's also not that hard (it's all really mechanical use of basic technique, although it is a little long).

But I was actually meaning the specific question "find the sum of the first n terms of k/2^k", which is quite easy.

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: March 28, 2011
Today on TSR

### University open days

1. University of Cambridge
Wed, 26 Sep '18
2. Norwich University of the Arts
Fri, 28 Sep '18
3. Edge Hill University
Faculty of Health and Social Care Undergraduate
Sat, 29 Sep '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams