# Tagged Questions

For questions on infinite products: convergence, computation, etc...

18 views

### Product of Uncountably Infinite Number of 1s

Just like the title says: What is the product of an uncountable number of 1s? Intuitively the answer is 1, but how does one go about defining such a product in general?
24 views

### Infinite product leading to $1/(1-z)$

Please give me a hint (i am studying Complex Variables for Engineering) on how to prove that ...
27 views

### Calculate the limits of sum and products [on hold]

Calculate the limits of: $$u_n= \prod_{i=1}^n (1+\frac{i}{n})$$ and $$u_n =\sum_{k=0}^n (C^k_n)^{-1}$$ Thank you a lot. This I my first time posting questions. Tell me if there is something ...
54 views
+100

### Pre measure for an infinite product of measure spaces

Let $\{(\Omega_k, \Sigma_k, P_k)\}_{k\geq 1}$ be a sequence of probability spaces. I am trying to prove the statement below in order to use it and get a pre measure and then use the Hahn kolomogrov ...
24 views

### Show that a martingale is not $L^1$ convergent

Consider the symmetric random walk $S_n$ on $\mathbb{Z}$. The process $Z_n=\exp(uS_n-n \ \log(\cosh(u)))$ for $u\in \mathbb{R}$ is a positive martingale with $E(Z_n)=1$ for all $n\geq 1$. $Z_n$ is ...
28 views

### Calculation of infinite product

My question is to prove the identity: $$\prod_{n=1}^{\infty}(\frac{\cos t-1}{n}+1)=\exp\bigg(-\int_0^1x^{-1}(1-\cos xt)dx\bigg)$$ which arises as a product of characteristic functions of a sequence ...
81 views

264 views

### Is $\prod_{n=1}^\infty P_{2n-1}$ regularizable?

Assume that $P_n$ denotes the $n$'th prime for this entire question. Inspriation: I was dumbfounded by the fact that: $$\hat\prod_\limits{n=1}^\infty P_{n}=4\pi^2$$ After further investigation, I ...
64 views

28 views

### Stuck in proof of Lemma 5.8 in Conway's Functions of one complex variable I

5.8 Lemma Let (X,d) be a compact metric space and let $\{g_n\}$ be a sequence of continuous functions from X into $\mathbb{C}$ such that $\sum g_n(x)$ converges absolutely and uniformly for x in X. ...
49 views

### How do I compute $\prod_{n=1}^\infty \mathrm{e}^{{\mathrm{i}\pi}/{2^n}}$?

I'm struggling with how to compute the following product: $$\prod_{n=1}^\infty \mathrm{e}^{{\mathrm{i}\pi}/{2^n}}$$ Wolfram Alpha tells me it's $-1$, and I can confirm that it converges since ...
32 views

64 views

### If $\prod_{n}(1+a_n)$ converges, does $\sum_{n}\frac{a_n}{1+a_n}$ converge?
I have a sequence of complex numbers $a_1,a_2,...$ such that $a_i \neq -1$. I then have the infinite product $\prod_{n=1}^{\infty}(1+a_n)$ which I know converges to a non zero complex number. I was ...