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.

  • 1
    $\begingroup$ 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. $\endgroup$ – Brian M. Scott Jul 23 '13 at 3:53
  • $\begingroup$ Could you define "meaningful" and "natural"? $\endgroup$ – Pedro Tamaroff Jul 23 '13 at 4:03
  • $\begingroup$ @PeterTamaroff In history, sometimes we first got some special cases, and then develop theory. Some special cases just have some special values. $\endgroup$ – User19912312 Jul 23 '13 at 4:05
  • $\begingroup$ @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 $\pi$? $\endgroup$ – Pedro Tamaroff Jul 23 '13 at 4:19
  • $\begingroup$ @PeterTamaroff no, sums to 1 is expected, sums to $\pi$ is an example to show some historical meanings. $\endgroup$ – User19912312 Jul 23 '13 at 4:28

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.


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