1
vote
1answer
43 views

Limit of the “productory”

With the term "productory" I just mean $\Pi_{i=m}^nx_i$ but I do not know the english term. My question is: is there a limit for such an expression in the same sense as the limit of a sum is an ...
6
votes
4answers
121 views

What does $\prod_{n\geq2}\frac{n^4-1}{n^4+1}$ converge to?

What does $\prod_{n\geq2}\frac{n^4-1}{n^4+1}$ converge to? As far as I can tell, this has no closed-form solution (not saying much, I don't know much math), but a friend of mine swears he saw a ...
0
votes
2answers
54 views

Let $a > 0$ and let $k \in \mathbb{N}$. Evaluate the limit

Let $a > 0$ and let $k \in \mathbb{N}$. Evaluate: $$\lim_{n\to \infty}a^{-nk}\prod ^k_{j=1}\left(a+\frac{j}{n}\right)^n$$ Clueless on this problem. Seek your help.
1
vote
3answers
72 views

Finding limit of a product.

Prove:$$\lim_{n \to\infty }\frac{1}{n}\left[\prod_{i=1}^{n}(n+i) \right ]^{\frac{1}{n}}=\frac{4}{e}$$ I tried using Squeeze Theorem but can't go beyond $1<L<2$. $$\lim_{n\to\infty} \left( 1 + ...
2
votes
2answers
96 views

Finding $\lim_{n\to\infty}\frac{\prod_{k=1}^n(2k-1)}{(2n)^n}$

Recently got this on a test: $$\lim_{n\to\infty}\frac{\prod_{k=1}^n(2k-1)}{(2n)^n}$$ Because it's a freshman calculus course, I think we were expected to solve it like a physicist. Taking a look at ...
5
votes
5answers
2k views

Can the limit of a product exist if neither of its factors exist?

Show an example where neither $\lim\limits_{x\to c} f(x)$ or $\lim\limits_{x\to c} g(x)$ exists but $\lim\limits_{x\to c} f(x)g(x)$ exists. Sorry if this seems elementary, I have just started my ...
14
votes
5answers
700 views

Evaluating the infinite product $\prod_{k=2}^{\infty }\left ( 1-\frac{1}{k^2} \right ).$

Evaluate $$\lim_{ n\rightarrow\infty }\prod_{k=2}^{n}\left ( 1-\frac{1}{k^2} \right ).$$ I can't see anything in this limit , so help me please.
2
votes
1answer
114 views

Limit of an n-ary product

Since a definite integral is defined as $$\lim_{n\to\infty} \sum_{i=0}^n f(x_i^*)\,\Delta x = \int_a^b f(x)\,dx$$ and the integral is much easier to calcluate than a sum, if we change the sum to a ...
2
votes
2answers
207 views

Approach to limit of infinite product

I was wondering if there is any proof that the limit of infinite product $$\lim_{n \to \infty} \prod_{i=1}^{n} x_i, \mathrm{where}$$ $$0 < x_i < 1$$ is equal to 0 and that it does not ...
3
votes
2answers
88 views

Calculate $\lim_{n\to\infty}[(1+x)(1+x^2)(1+x^4)\cdot\cdot\cdot(1+x^{2n})]$, $|x|<1$

Please help me solving $\lim_{n\to\infty}[(1+x)(1+x^2)(1+x^4)\cdot\cdot\cdot(1+x^{2n})]$, $|x|<1$
4
votes
2answers
595 views

How to evaluate $\lim\limits_{n\to+\infty} \prod\limits_{k=1}^n (1+k/n^2)$?

I've got a limit which puzzle me several days. The question is $$ \lim_{n\to+\infty} \prod_{k=1}^n\left(1+\frac{k}{n^2}\right).$$ Can you help me? Thank you in advance
21
votes
2answers
600 views

Compute $\lim\limits_{n\to\infty} \prod\limits_2^n \left(1-\frac1{k^3}\right)$

I've just worked out the limit $\lim\limits_{n\to\infty} \prod\limits_{2}^{n} \left(1-\frac{1}{k^2}\right)$ that is simply solved, and the result is $\frac{1}{2}$. After that, I thought of calculating ...
4
votes
5answers
856 views

The limit of infinite product

Is there any elementary proof that the limit of infinite product $\prod_{k=1}^{\infty}(1-\frac{1}{2^k})$ is not 0?