isi-entrance None Q5

isi-entrance · India · subjective_collection Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums
Evaluate $\lim_{n \rightarrow \infty} \left\{\frac{1}{n+1} + \frac{1}{n+2} + \ldots + \frac{1}{n+n}\right\}$.
$$\lim_{n \rightarrow \infty} \frac{1}{n}\left[\frac{1}{1+\frac{1}{n}} + \frac{1}{1+\frac{2}{n}} + \cdots + \frac{1}{1+\frac{n}{n}}\right] = \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^{n} \frac{1}{1+\frac{r}{n}} = \int_0^1 \frac{dx}{1+x} = \left[\log_e(1+x)\right]_0^1 = \log_e 2$$ 
Evaluate $\lim_{n \rightarrow \infty} \left\{\frac{1}{n+1} + \frac{1}{n+2} + \ldots + \frac{1}{n+n}\right\}$.
Paper Questions
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