Question

The value of \[ \lim_{n \to \infty} \left\{ \frac{1}{n+m} + \frac{1}{n+2m} + \frac{1}{n+3m} + \dots + \frac{1}{n+nm} \right\} \] is:

• A. $\dfrac{\log_e(m)}{m}$
• B. $\dfrac{\log_e(1+m)}{1+m}$
• C. $\dfrac{\log_e(1+m)}{m}$
• D. $\dfrac{\log_e(1+m)}{1-m}$

Hint:

Whenever a limit contains a summation with terms involving: \[ \frac{r}{n} \] try converting it into a Riemann integral using: \[ \frac{1}{n}\sum f\left(\frac{r}{n}\right) \rightarrow \int_0^1 f(x)\,dx \] Factoring out $n$ from the denominator is usually the key first step.

Answer 1

The Correct Option is C