Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Give me some integral and series whose value is $1$.

Where can I find a large number of these kinds of examples.

I have two examples here, but I cannot think up more...

This is geometry series, we learned in high school and before.

$$\begin{align*}\sum _{i=1}^{\infty } 2^{-i}=1\end{align*}$$

This is from $\text{Central}\text{ }\text{Limit}\text{ }\text{Theorem}$.

$$\begin{align*}\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{\infty } e^{\frac{-x^2}{2}} \, dx=1\end{align*}$$

I'd like some more historical /meaningful/natural/interesting examples.

Something like some series whose value is $\pi /4$ or $\pi$,

$$\begin{align*}\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\text{...}=\frac{\pi }{4}\end{align*}$$

In history, sometimes we firstly got some special cases with some special values, and then we continue developing theories.

Here my question is about that sums to $1$, and that sums to $\pi$ is only an example to show some historical meanings.

share|cite|improve this question
Every convergent series with a non-zero sum trivially gives you an example: if $\sum_{n\ge 0}a_n=L$, then $\sum_{n\ge 0}b_n=1$, where $b_n=\frac{a_n}L$. A similar trick works for the integrals. – Brian M. Scott Jul 23 '13 at 3:53
Could you define "meaningful" and "natural"? – Pedro Tamaroff Jul 23 '13 at 4:03
@PeterTamaroff In history, sometimes we first got some special cases, and then develop theory. Some special cases just have some special values. – User19912312 Jul 23 '13 at 4:05
@spuorg-imes I don't understand. You wanted a series that sums to $1$, there you go. Now you want a series that sums to $\pi$? – Pedro Tamaroff Jul 23 '13 at 4:19
@PeterTamaroff no, sums to 1 is expected, sums to $\pi$ is an example to show some historical meanings. – User19912312 Jul 23 '13 at 4:28
up vote 6 down vote accepted

Take you favorite convergent series $$\sum a_n=A\neq 0$$ Define $$a_n^\prime=\frac{a_n}A$$

The same for the integral. This procedure is similar to what we usually know as normalization.

share|cite|improve this answer
+1: Same as my answer here – user17762 Jul 23 '13 at 3:59

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.