Series of functions & uniform convergence

[kingwinner]

Homework Statement

Homework Equations

The Attempt at a Solution

This is not graded homework, but optional exercises I found in my textbook. It's days before my exam, but I'm still not sure how to do problems like this. I would really appreciate if someone can teach me how to solve this.

For part a, I think we have to look at the partial sum, but how can we get the partial sum in the first place?

For part b, should we use the Weierstrass M-test? Is it true that to prove that a SERIES of functions is uniformly convergent, most of the time we're going to use the Weierstrass M-test? Is this the only way?

Thanks a lot!

 

[Office_Shredder]

For part (a), factor the x² in the numerator out of the series and you're left with a geometric serie

 

[kingwinner]

a) OK, so S(x) = x² / [1- 1/(1+x²)], is this correct?
But how can we prove that |1/(1+x²)|<1? I don't think it's always true, e.g. what if x=0?

Thanks.

 

[Office_Shredder]

If x is not zero, 1+x²>1 and you should be able to handle that case.

If x=0, what are the terms you're summing up? They're all zero, so you don't even need to use the geometric series for that one

 

[kingwinner]

OK, so the answer for part a is:
S(x) = x² / [1- 1/(1+x²)] = 1+x² if x≠0
S(x) = 0 if x=0


Can someone help me with the harder part (i.e. part b), please? In general, I am having a lot of headaches on problems about uniform convergence. I know the precise definition of it, and I re-read the definition many times, but I have no idea how to actually APPLY the definition to solve actual problems.
Should we use the Weierstrass M-test here? Is this the only way to prove that a SERIES of functions is uniformly convergent?

Thanks for any help!

 

[rs1n]

OK, so the answer for part a is:
S(x) = x² / [1- 1/(1+x²)] = 1+x² if x≠0
S(x) = 0 if x=0


Can someone help me with the harder part (i.e. part b), please? In general, I am having a lot of headaches on problems about uniform convergence. I know the precise definition of it, and I re-read the definition many times, but I have no idea how to actually APPLY the definition to solve actual problems.
Should we use the Weierstrass M-test here? Is this the only way to prove that a SERIES of functions is uniformly convergent?

Thanks for any help!

The definition I prefer to use says that $S_k \to S$ uniformly on E if

$$\lim_{k\to\infty} \left| S_k(x) - S(x) \right| = 0$$

for all $x\in E$. In epsilon notation, we should be able to find an N (independent of any particular value of x) such that

$$\left| S_k(x) - S(x) \right| < \epsilon$$

for all $x\in E$ and all $k>N$. One approach to showing uniform convergence is to estimate the difference within the absolute value signs and somehow get an upper bound on it -- with the upper bound not containing any x's (and instead, containing only constants and n).

 

[kingwinner]

What is Sk? Is it the partial sum?

I get the idea and understand the definition, but I just don't know how to prove it. Can you do an example on some interval [a,b] to prove uniform convergence or nonconvergence?

Thanks!

 

[Char. Limit]

What is Sk? Is it the partial sum?

Yes.

I get the idea and understand the definition, but I just don't know how to prove it. Can you do an example on some interval [a,b] to prove uniform convergence or nonconvergence?

Well... for what x does this converge? Here's a hint... after you factored the x^2 out, got the geometric series, and attempted to evaluate the interval of convergence, what was your answer for all x not equal to 0?

Thanks!

No problem.

 

[kingwinner]

Sorry, I don't quite get what you're saying...:(

So how can we prove part b??

 

[Char. Limit]

Start with a limit test...

What is the limit of $$\frac{x^2}{(1+x^2)^n}$$ as n goes to infinity, if x doesn't equal zero?

 

[kingwinner]

Start with a limit test...

What is the limit of $$\frac{x^2}{(1+x^2)^n}$$ as n goes to infinity, if x doesn't equal zero?

I think the limit is 0 (if x=/=0). But how is this going to help?

 

[Char. Limit]

I think the limit is 0 (if x=/=0). But how is this going to help?

Sorry, I shouldn't have said limit test... my bad.

Do a ratio test instead... and check for which x is the series convergent or divergent.

 

[kingwinner]

No, I don't think it's going to work. The ratio test only tells you whether it is convergent or divergent.

But the question asks for UNIFORM convergence/nonconvergence, that's the key point, right?

Can someone help me with part b, please?

 

[Gib Z]

1) Compute $S_k (x)$
2) Plug that into the LHS of the limit definition of Uniform Convergence in post 6 by r1sn.
3) Try and evaluate the limit and if you can't, show us exactly where you run into problems in doing so.

 

[kingwinner]

1) Compute $S_k (x)$
2) Plug that into the LHS of the limit definition of Uniform Convergence in post 6 by r1sn.
3) Try and evaluate the limit and if you can't, show us exactly where you run into problems in doing so.

How can we compute $S_k (x)$? In this case we're only summing over a finite number of terms. How can we use the formula for geometric series?

Also, is there any way to use the "Weierstrass M-test" in this problem to prove uniform convergence, instead of going back to the basic definitions?

Also, how can I know what interval [a,b] to work with to prove uniform convergence?

Thanks for answering!

 

[Gib Z]

How can we compute $S_k (x)$? In this case we're only summing over a finite number of terms. How can we use the formula for geometric series?

We instead use the formula for a finite geometric sum: $$\sum_{n=0}^k ar^n= \frac{a(1-r^{k+1})}{1-r}$$.

Also, is there any way to use the "Weierstrass M-test" in this problem, instead of going back to the basic definitions?

Yes there is, but it is not the easiest way to do the problem, as I just realized.
One can find the Geometric sum and proceed via the definition presented by r1sn, or find some suitable constant with respect to x to apply the M-test with, but I found an easier way:

Case 1: x=0, The sum is trivially zero.
Case 2: |x|>0, Then add and subtract 1 in the numerator of f_n (x). Then S_n(x) becomes a telescoping sum.

Also, how can I know what interval [a,b] to work with to prove uniform convergence?

Thanks for answering!

Basically by inspection, seeing which values of a and b allows our proof of uniform convergence to be valid.

 

[kingwinner]

Basically by inspection, seeing which values of a and b allows our proof of uniform convergence to be valid.

How can we tell which intervals give uniform convergence and which intervals does not give uniform convergence just by inspecting?

 

[Gib Z]

In a similar way to how we recognize that eg $$\sum 1/k^p$$ converges if p>1. We see which part precisely determines the property we want and think about what values give what.

Don't inspect the expression for the function or summation, look at the expression for the limit of |s_k (x) - s(x)| instead.

 

[kingwinner]

Since the pointwise limit function is discontinuous, I think any interval that contains 0 would not give uniform convergence. But are there any other restrictions?

For example, do we have uniform convergence on [1,a] for any a>1? If so, how can we prove it by the Weierstrass M-test (or definition)? Can someone show an example on some particular interval? I just don't see how the proof works out.

I hope someone can help me out! Thank you!