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Pattern – 26 : Definite integration

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⋆ Pattern – 26

Method to express infinite series as define integrals.

Step – 1 :-

Find $T_r=r^{th}$ term of infinite series.

Step – 2 :-

Taking common $\frac{1}{n}$ from each terms of Tr (for r = 1 to n) & convert the terms except common term in form of $\frac{r}{n}$ .

i.e. $T_r=\frac{1}{n}\ f\left(\ \frac{r}{n}\ \right)$

Step – 3 :-

Now, that infinite series = $S_\infty$ becomes

$S_\infty\ \ \ =\ \ \begin{matrix}\lim\\n\rightarrow\infty\\\end{matrix}\sum_{r\ =\ \ 1}^{p\ n}{\frac{1}{n}\ f\left(\frac{r}{n}\right)}$

We will definitely get 1st term by taking r = 1 & will get nth term by taking r = pn

So limits of summation is r = 1 to r = pn

Step – 4 :-

Now, in $\sum_{r\ =\ \ 1}^{p\ n}{\frac{1}{n}\ f\left(\frac{r}{n}\right)}$

Replace $\frac{r}{n}$ by x, $\frac{1}{n}\ by \ dx$

Lower Limit, $\alpha = \lim_{n \rightarrow \infty } \frac{r}{n}$

$= \lim_{n \rightarrow \infty } \frac{1}{n}$

= 0

Upper Limit, $\beta\ =\ \lim_{n\ \rightarrow\ \infty}{\ \frac{r}{n}}$

$=\ \lim_{n\ \rightarrow\ \infty}{\ \ \frac{pn}{n}}\ \$

= p

Step – 5 :-

We will get,

$S_\infty\ \ \ =\ \ \begin{matrix}\lim\\n\rightarrow\infty\\\end{matrix}\sum_{r\ =\ \ 1}^{p\ n}{\frac{1}{n}\ f\left(\frac{r}{n}\right)}$

$=\ \int_{0}^{p}\ f\left(x\right)\ \ dx$