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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