Intereting Posts

Can one differentiate an infinite sum?
Closed form for $f(z)^2 + f ' (z)^2 + f ' ' (z) ^2 = 1 $?
Divergent or not series?
Does an equivalence of $G$-sets and $H$-sets imply an isomorphism of $G$ and $H$?
Constructing a number not in $\bigcup\limits_{k=1}^{\infty} (q_k-\frac{\epsilon}{2^k},q_k+\frac{\epsilon}{2^k})$
Prove that largest root of $Q_k(x)$ is greater than that of $Q_j(x)$ for $k>j$.
Hanging a picture on the wall using two nails in such a way that removing any nail makes the picture fall down
How to evaluate $I=\displaystyle\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)\ \mathrm dx$
Overlapping Probability in Minesweeper
Closed form for n-th anti-derivative of $\log x$
Show that this entire function is polynomial.
What axioms does ZF have, exactly?
Bijection between an infinite set and its union of a countably infinite set
Question about Generalized Continuum Hypothesis
Can compact sets completey determine a topology?

What is the current state of the art in summing (where by ‘summing’, I mean ‘representing in terms of already known constants and whatnot’) series such as these:

$$\sum_{n=1}^{\infty} \frac{1}{\sqrt[3]{1+7^{n}}}$$

$$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n!}}$$

- Does $\sum_{n\ge1} \sin (\pi \sqrt{n^2+1}) $ converge/diverge?
- Closed set in $\ell^1$
- Borel-Cantelli-related exercise: Show that $\sum_{n=1}^{\infty} p_n < 1 \implies \prod_{n=1}^{\infty} (1-p_n) \geq 1- S$.
- $\sum_{k=1}^n(k!)(k^2+k+1)$ for $n=1,2,3…$ and obtain an expression in terms of $n$
- Evaluate $\int_{0}^{+\infty }{\left( \frac{x}{{{\text{e}}^{x}}-{{\text{e}}^{-x}}}-\frac{1}{2} \right)\frac{1}{{{x}^{2}}}\text{d}x}$
- Any converging sequence is bounded

$$\sum_{n=1}^{\infty} e^{-\sqrt{n}}$$

$$\sum_{n=1}^{\infty} \frac{1}{n^{3}+\sqrt[7]{n}}$$

I have a copy of Konrad Knopp’s book /Theory and Application of Infinite Series/ , but that’s fifty years old, and I’ve been hoping that there have been improvements in the techniques since then.

- Is there a series to show $22\pi^4>2143\,$?
- When are two series the same?
- Looking for strictly increasing integer sequences whose gaps between consecutive elements are “pseudorandom”
- Existence of increasing sequence $\{x_n\}\subset S$ with $\lim_{n\to\infty}x_n=\sup S$
- Sum of $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{2n-1}}{2n-1}$
- Another “can't use Alt. Series Conv. test” (How do I determine if it converges or diverges?)
- $\sum_1^n 2\sqrt{n} - \sqrt{n-1} - \sqrt{n+1} $ converge or not?
- Two sums with Fibonacci numbers
- Proving this formula $1+\sum_{n=0}^{\infty }\frac{1}{\pi \left(2n+\frac{3}{4}\right)\left(2n+\frac{5}{4}\right)}=\sqrt2$
- Evaluate $\lim \limits_{n\rightarrow\infty}\sin^2(\pi\sqrt{n^2 + n})$

Well, much depends on the individual series but one interesting development in the last 50 years is on the acceleration of convergence, which enables us to express some slowly convergent series in terms of one(s) with much faster convergence, or a sum of known constants plus faster converging series. For example, I really like this transformation due to Simon Plouffe

$$\zeta(7) = \frac{19}{56700}\pi^7 – 2\sum_{n=1}^{\infty} \frac{1}{n^7(e^{2\pi n}-1)},$$

where $\zeta(7) = \sum_ {n=1}^{\infty} 1/n^7,$ which is given on this wikipedia page.

Andrei Markov’s 1890 method, which was the basis for Apéry’s proof of the irrationality of $\zeta(3),$ has been pushed further by Mohammed and Zeilbeger.

You can see some other nice transformations of $\zeta(3)$ here.

You may want to read up on Gosper’s algorithm. It helps you find closed-form expressions for certain classes of sums.

Take a look at Petkovsek, Wilf and Zeilberger’s book “A = B”. They tame an entire zoo of sums. Also of interest is Wilf’s “generatingfunctionology”, his “snake oil method” is simple to apply by hand and works in a surprising variety of situations.

- Gradient of a vector field?
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- Left Inverse: An Analysis on Injectivity
- Is $6.12345678910111213141516171819202122\ldots$ transcendental?
- Real Analysis Boundedness of continuous function
- $\sqrt{A(ABCD)} =\sqrt{A(ABE)}+ \sqrt{A(CDE)}$
- How to prove the distributive property of cross product
- Ideal class group of $\mathbb{Q}(\sqrt{-65})$
- Inner Product Spaces over Finite Fields
- Certain step in the induction proof $\sum\limits_{i=0}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$ unclear
- This sigma to binom?
- A question about associativity in monoids.
- ALL Orthogonality preserving linear maps from $\mathbb R^n$ to $\mathbb R^n$?
- Isosceles triangle
- Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $$