Sequence of functions uniformly/absolute/normal convergence

Question

I've got this sequence of functions...

$f(x) = \sum_{n=0}^{\infty} \frac{x^2}{(1+x^2)^n}$

To show:

1. The sequence is convergent in $\mathbb{R}$
2. It is NOT uniformly convergent in compact intervals [a,b] that contain zero.

Additional exercise:

3. In which compact intervals is the sequence uniformly convergent
4. In which compact intervals is the sequence NOT normal convergent
5. Calculate the sum-function of the sequence

What I got:

First I tried to calculate the limit (in form of a function) to which the sequence converges. In this case (because it's a geometric sequence) the limit is

$f(x) = lim_{n\rightarrow \infty} f_n(x) = \frac{x^2}{1-\frac{1}{1+x^2}}=1+x^2$

From what i can determine no interval [a,b] with $a<0<b \vee a,b = 0$ can be uniformly convergent because $\forall n\in \mathbb{R},x=0 : \frac{0^2}{(1+0^2)^n}=0$

Therefore the limit in this point must be zero and can never be, as the limit function suggests, be 1 ($1+0^2=1$)

So is it right to deduct that every other interval is uniformly convergent? At least it seems like it, is there a way to check?

(2) check

Next i tried to show that the sequence of functions is absolute convergent by using the ratio test. It gives me $lim_{n\rightarrow \infty} |\frac{1}{1+x^2}| < 1$ which is true for all x except x = 0.

The question states otherwise though. It must be true for ALL x.

(1) check?

(3) is just like (2) somehow. The answer: For all intervals without x.

(3) check?

I've no clue for (4) :(

(4) OPEN

For the last one: The Sum-function. It you're not familiar with it, it is just a function that lets you get the value of the sequence ASAP. E.G. $f(x) = \sum_{n=0}^{M} n = n*\frac{n+1}{2}$

In my case, it's just the limit-function isn't it?

(5) check

Do you have any helpful recommendations, helpful clues or improvements in mind that would help me?

Best regards.

Answer 1

I think you would be better if you write $$f_N(x)=\sum_{n=0}^{N}{x^2\over (1+x^2)^n}=(1+x^2)^{N+1}-1$$and $$f(x)=\lim_{N\to \infty}f_N(x)$$the your arguments become preciser and easier.

Hint for $(4)$

By differentiation we obtain that the maximum of $x^2\over (1+x^2)^n$ is $${(n-1)^{n-1}\over n^n}\sim{1\over e}\cdot {1\over n-1}$$so the question is: what happens if an interval contains $0$ or does not contain zero?