# Some nice theorem to find the value of serie

I know a lot of techniques to show that a serie converge or diverge (Cauchy condensation test, Abel's test, comparision with an integrale, root test, ratio test...).

On the other hand I know nothing about caculating the value of a serie. Is there any theorems that may help ?

From now on I know the following :

• integrale test can be combine with squeeze theorem
• geometric serie

For example in this [How to prove $\lim \limits_{x \to 1^-} \sum\limits_{n=0}^\infty (-1)^nx^{n²} = \frac{1}{2} \$? there are talking about :

• Herglotz' trick
• Weierstrass products.

• ...

So I would like to know if there are some other nice theorem or slick techniques to find the value of a serie.

Thank you !

## 2 Answers

In this regard, series are much like integrals: you need knowledge and, even more, ingenuity and experience.

Some basic Theorems, besides those you cited, are:
- Indefinite Sum;
- Summation by parts; - (Discrete) Convolution;
- Partial Fractions Decomposition;
- Binomial Tranform and others;
- Gosper's Algorithm and Hypergeometric Function;
- Abel's Binomial Theorem
and many others

• +1 for Gosper's Algorithm - found this a great presentation of why it works for WZ method: people.brandeis.edu/~gessel/homepage/slides/wilf80-slides.pdf @auhasard well word a read if you have a chance. Came across the method in a nice compendium called Irresistible Integrals think this is a free pdf: researchgate.net/publication/… plenty of material in here! – Mehness Jun 9 '18 at 16:39
• (hoho pls ignore the bogus link, this is the book I meant if interested: Irresistible Integrals, Symbolics, Experiments, Evaluation – Mehness Jun 9 '18 at 21:17
• @Mehness: thanks for indicating these interesting publications. I'll try and get through them. – G Cab Jun 9 '18 at 23:39
• No problem sir, the book is definitely just sth to dip into if interested it's pretty hefty! – Mehness Jun 9 '18 at 23:48
• @GCab Thank you ! This is actually what I was looking for, now it's time to train ! – auhasard Jun 10 '18 at 8:31

I don't know if this is a good approach and if you mentioned this but it would add another possible approach which I posted on the linked question.

Consider

$$F(y+1)-F(y) = e^{i\pi y} x^{y^2} = e^{i \pi y -ay^2} = f,$$

Take fourier transform of this and solve e.g. $$\hat F(s) = \frac{\hat f(s)}{e^{is} - 1}$$ Note that $$\lim_{n \to \infty} \sum_{i=0}^{n} F(i+1)-F(i) = - F(0) = \lim_{n \to \infty} F(n+1) - c \int \mathcal{F}(f)(s)\,ds = - c \int \mathcal{F}(f)(s)\,ds,$$ under the assumption $\lim_{n\to\infty} F(n+1) = 0$

For the example above it turns out the fourier transform turns into a delta measure as we approach $x\to 1$ and this might be an approach that is tractable and lead to convergence proof and limit calculation.

• I think you meant $n=\infty$ in your sum. Otherwise there's an extra term $F(n+1)$, . – Alex R. Jun 8 '18 at 23:32
• thanks! I'll fix it – Stefan Jun 11 '18 at 15:09