1
vote
0answers
41 views

Are there such prime giving functions?

Here let us define a function $f : \mathbb{N} \rightarrow \mathbb{N}$ , such that for every $n$ , The sequence $\{f(n) ,f(n)+1 ,f(n)+2 , f(n)+3, \dots , f(n)+n\}$ contains atleast $1$ prime . Let us ...
2
votes
0answers
213 views

List of Common or Useful Limits of Sequences and Series

There are many sequences or series which come up frequently, and it's good to have a directory of the most commonly used or useful ones. I'll start out with some. Proofs are not required. ...
6
votes
3answers
200 views

What is the fastest way to $\pi$?

There are many known sequences convergent to $\pi$ with different convergence accelerations. For example both of the following sequences are convergent to $\pi$ when $n$ goes to $\infty$: (a) ...
37
votes
3answers
655 views

Conjectural closed-form representations of sums, products or integrals

What are some examples of infinite sums, products or definite integrals that have conjectural closed form representations that were confirmed by numerical calculations up to whatever maximum precision ...
0
votes
2answers
425 views

List Table(s) of Series Here

I've been interested in series expansions of all types of mathematical functions. I was wondering if anyone has ever created a large list of all types of series. For example, Wolfram's Mathworld's ...
1
vote
0answers
61 views

A gratifying re-encounter with a piece of math that was out of my mind

A series of real numbers is said to be conditionally convergent if it is convergent but not absolutely convergent. By rearranging the terms of a conditionally convergent series we can make the ...
4
votes
6answers
544 views

What is some infinite series expansion for 3/7?

What is some infinite series expansion for 3/7? and so on for these fraction with digit in base 10? I can't think of some useful thing at all. Please generalize some useful series expression for all ...
1
vote
2answers
87 views

What is the list of theorem that are able to find out a sequence is converge or not?

A sequence is called converge if for every next term of the sequence is getting closer to the limit of a number. What is the list of theorem that are able to helping to find out a sequence is converge ...
3
votes
1answer
439 views

Uses of Divergent Series and their summation-values in mathematics?

When I was trying to find closed-form representations for odd zeta-values, I used $$ \Gamma(z) = \frac{e^{-\gamma \cdot z}}{z} \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{-1} e^{\frac{z}{n}} $$ ...
216
votes
21answers
19k views

Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$

As I have heard people did not trust Euler when he first discovered the formula $$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs. I ...
4
votes
2answers
146 views

What is an alternative formulation to a contour integral?

Suppose that a rational generating function is given representing the function $f(x) = \sum_{i=0}^{2n}{c_i x^i}$ where $c_i \in \mathbb{N}$ The goal is to determine if the coefficient in the middle, ...