Proof that $E[X^2]$ = $\sum_{n=1}^\infty (2n-1) P(X\ge n)$

X is a random variable with values from $ \Bbb N\setminus{0}$

I am trying to show that $ E[X^2]$ = $ \sum_{n=1}^\infty (2n-1) P(X\ge n)$ iff $ E[X^2]$ < $ \infty$ .

I rewrote $ P(X \ge n)$ :

$ E[X^2]$ = $ \sum_{n=1}^\infty (2n-1)\sum_{x=1}^\infty 1_{x \ge n}P(X=x)$

Now I tried to rearrange the sums:

$ E[X^2]$ = $ \sum_{x=1}^\infty \sum_{n=1}^x (2n-1)P(X=x)$

But I think that I made a mistake. Could you give me some hints?

Divergence of $\sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}$: need further clarification

I see the following answer but am unable to understand it. Is someone able to add more steps clarifying it further? BTW, I added a duplicate question because I couldn’t add a comment for the original answer. Not enough reputation. Thanks!

Is it possible to show that $ \sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}$ diverges?

$\forall a > 0$ $\sum_{n=1}^{\infty} f(na)$ is convergent. Prove that $\int_{0}^{\infty}f(x) dx$ is convergent.

Hi can you help me solve this exercise? Thanks. Let $ f: [0;+\infty) \to \mathbb{R}$ be nonnegative and continuous function. Suppose $ \forall a > 0$ $ \sum_{n=1}^{\infty} f(na)$ is convergent. Prove that $ \int_{0}^{\infty}f(x) dx$ is convergent. I tried to solve it by using the Riemann sum, but for fixed a it doesn’t work. I have no other ideas.

Show that the complex series $\sum_{n=1}^\infty \frac{z^n}{n}$ converges for $z=i$.

I am trying to show that the complex series $ \sum_{n=1}^\infty \frac{z^n}{n}$ converges for $ z=i$ . I have worked out in a separate part of the problem that the radius of convergence is $ 1$ , but now I am trying to show it by substituting in $ i$ as follows:

$ \sum_{n=1}^{\infty}\frac{i^n}{n}=\frac{i}{1}-\frac{1}{2}-\frac{i}{3}+\frac{1}{4}+\frac{i}{5}-\frac{1}{6}-\frac{i}{7}+\frac{1}{8}+…$

$ =\left(-\frac{1}{2}+\frac{1}{4}-\frac{1}{6}+\frac{1}{8}+…\right)+i\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+…\right)$

Here’s where I get stuck. I know that by splitting a summation into real and imaginary parts, if we can show that the real part converges and that the imaginary part converges, the complex series converges. So, assuming that nothing has gone wrong in my thinking up to this point, I need to somehow show that both of the above parts converge. How would I go about doing this, and what would the parts converge to?

If $\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} f_{n}(k) = \infty$ then $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty} f_{n}(k) = \infty$.

Suppose $ \{f_{n}\}_{n=1}^{\infty}$ be functions such that $ f_{n} : \Bbb{N} \rightarrow \Bbb{R}^{+}$ for each $ n$ .

I was trying to prove –

If $ \sum_{n=1}^{\infty} \sum_{k=1}^{\infty} f_{n}(k) = \infty$ then $ \sum_{k=1}^{\infty} \sum_{n=1}^{\infty} f_{n}(k) = \infty$ .

I can see that both the double summation series are equal by expanding the double summation.But I am trying to prove it ? any other thoughts?

Hm, as the order of summation are changed, is Uniform convergence likely to play any role here?

What condition on the variable $x$ ensures that the power series $\sum_{n=1}^{\infty} a_nx^n \in \mathbb{Q}$?

Suppose the series $ \sum_{n=1}^{\infty} a_n$ converges in $ \mathbb{Q}$ i.e., $ \sum_{n=1}^{\infty} a_n \in \mathbb{Q}$ , where $ a_n$ is arbitrary rational numbers. Consider the power series $ \sum_{n=1}^{\infty} a_nx^n$ . What condition on the variable $ x$ ensures that the power series $ \sum_{n=1}^{\infty} a_nx^n \in \mathbb{Q}$ ?

Answer:

For particular case,

$ a_n=\frac{1}{p^n}$ , for some prime $ p$ .

Then the series $ \sum_{n=1}^{\infty} a_n=\sum_{n=1}^{\infty} \frac{1}{p^n}$ converges in $ \mathbb{Q}$ and in fact $ \sum_{n=1}^{\infty} \frac{1}{p^n}=\frac{1}{p-1} \in \mathbb{Q}$ .

Next consider the corresponding power series $ \sum_{n=1}^{\infty} \frac{x^n}{p^n}$ , it has sum $ S(x)=\frac{x}{p-x}$ .

Now $ S(x) \in \mathbb{Q} \Rightarrow S(x)=r \in \mathbb{Q} \Rightarrow x=S^{-1}(r), \ r \in \mathbb{Q} $ .

This is particular case.

Does there exists any general theory solving my above question where $ a_n$ is arbitrary rational number but gives condition on the variable or argument $ x$ ?

This concept is needed in my research.

Please someone help