Tagged Questions
9
votes
3answers
370 views
Prove this product
How to prove this product?
$$\prod\limits_{k=2}^ n {\frac{k^2+k+1}{k^2-k+1}}=\frac{n^2+n+1}{3}$$
8
votes
2answers
174 views
How to find finite trigonometric products
I wonder how to prove ?
$$\prod_{k=1}^{n}\left(1+2\cos\frac{2\pi 3^k}{3^n+1} \right)=1$$
give me a tip
4
votes
1answer
108 views
Evaluate $\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$
Difficult question from some test somewhere (I forget).
$$\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$$
$x$ is, of course, an integer.
1
vote
1answer
59 views
Infinite Product is converges
I am adding this problem since it is interesting and valuable to be verified here:
Prove that the infinite product $\prod_{k=1}^{\infty}(1+u_k)$, wherein $u_k>0$, converges if ...
2
votes
1answer
69 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 ...
3
votes
2answers
78 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 $\lim_{n\to\infty}[(1+x)(1+x^2)(1+x^4)\cdot\cdot\cdot(1+x^{2n})]$, $|x|<1$
2
votes
1answer
97 views
Is there a relationship between products and integrals? Is there a way to convert a product into an integral?
I know that the Euler-Maclaurin formula establishes a relationship between sums and integrals, but is there some sort of formula that establishes a relationship between products and integrals? I don't ...
4
votes
2answers
234 views
Infinite product
How do I solve the infinite product of $$\prod_{n=2}^\infty\frac{n^3-1}{n^3+1}?$$
I know that I have to factorise to $$\frac{(n-1)(n^2+n+1)}{(n+1)(n^2-n+1)},$$
but how do I do the partial product?
...
21
votes
2answers
477 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
1answer
162 views
Is there a “continuous product”?
Is there a "continuous product" which is the limit of the discrete product $\Pi$, just like the integral $\int$ is the limit of summation $\sum$.
Thanks!
2
votes
1answer
93 views
Infinite product of recursive sequence
Let $a_{n+1}=\sqrt {(a_n+a_{n-1})/2}$ and $a_0=a_1=2$, how to prove convergence of the product $a_0 a_1 a_2 a_3...a_\infty$, and possibly find its value?
2
votes
2answers
472 views
The derivative of a product of more than two functions
I'm trying to generalize the product rule to more than the product of two functions using the fact that I can treat the product of $n$-1 functions as a single one. Here is an example of what I mean:
...
3
votes
2answers
54 views
interval for a product to infinity
I was wondering - how would I specify the interval (the amount that n increases each time) between terms? Is that possible? What if I want it to increase by, say, ...
