Definite Integration Question 6
Question 6
- Show that, $\lim _{n \rightarrow \infty} \frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{6 n}=\log 6$.
$(1981,2 \mathrm{M})$
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Solution:
- $\lim {n \rightarrow \infty} \frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{6 n}=\sum{r=1}^{5 n} \frac{1}{n+r}$
$$ \begin{aligned} & =\lim {n \rightarrow \infty} \frac{1}{n} \sum{r=1}^{5 n} \frac{1}{1+\frac{r}{n}} \ & =\int_{0}^{5} \frac{d x}{1+x}=[\log (1+x)]_{0}^{5}=\log 6-\log 1=\log 6 \end{aligned} $$
Here,
