# JEE Main & Advanced Mathematics Definite Integrals Summation of Series by Integration

Summation of Series by Integration

Category : JEE Main & Advanced

We know that $\int_{a}^{b}{f(x)dx}=\underset{n\to \infty }{\mathop{\lim }}\,h\sum\limits_{r=1}^{n}{f(a+rh)}$, where $nh=b-a$

Now, put  $a=0,$ $b=1,$ $\therefore$ $nh=1$ or $h=\frac{1}{n}$.

Hence $\int_{0}^{1}{f(x)\,dx=\underset{n\to \infty }{\mathop{\lim }}\,}\frac{1}{n}\sum\limits_{{}}^{{}}{f\left( \frac{r}{n} \right)}$.

Express the given series in the form $\sum{\frac{1}{n}f\left( \frac{r}{h} \right)}$.

Replace $\frac{r}{n}$by $x,\,\,\frac{1}{n}$ by $dx$ and the limit of the sum is $\int_{0}^{1}{{}}f(x)\,dx$.

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