JEE WhatsApp Group
Sign in
Please select an account to continue using cracku.in
↓ →
JEE WhatsApp Group
Join Our JEE Preparation Group
Prep with like-minded aspirants; Get access to free daily tests and study material.
$$\lim_{n \to \infty} \left(\frac{1}{1+n} + \frac{1}{2+n} + \frac{1}{3+n} + \ldots + \frac{1}{2n}\right)$$ is equal to :-
$$\lim_{n\to\infty} \frac{1}{n} \sum_{r=1}^{n} f\left(\frac{r}{n}\right) = \int_{0}^{1} f(x) \, dx$$
$$\lim_{n\to\infty} \sum_{r=1}^{n} \frac{1}{n+r} = \lim_{n\to\infty} \frac{1}{n} \sum_{r=1}^{n} \frac{1}{1 + \frac{r}{n}}$$
$$\implies\int_{0}^{1} \frac{1}{1+x} \, dx$$
$$\left[ \log_e(1+x) \right]_{0}^{1} = \log_e 2 - \log_e 1 = \log_e 2$$
Educational materials for JEE preparation
Ask our AI anything
AI can make mistakes. Please verify important information.
AI can make mistakes. Please verify important information.