Question

Check whether the sequence { an} , where an= 1/(n+1)+ 1/(n+2)+..1/(2n) is convergent or not.

Answer 1

Solution. The terms $\frac{1}{2n}$ are positive and decreasing, and since $\lim_{n \to \infty} \frac{1}{2n} =0$. The necessary criterion for the convergence of the sequence is satisfied.

Let's check the convergence of the sequence using the integral test (either both the integral and the series converge, or both diverge).

Find $\int_1^\infty \frac{dx}{2x}$

$$\int_1^\infty \frac{dx}{2x}= \lim_{t \to \infty} \int_1^t \frac{dx}{2x}= \lim_{t \to \infty} \frac {1}{2}ln(x) \mid_1^t=$$

$$=\lim_{t \to \infty} \frac {1}{2}(ln(t) -(ln(1)) = \lim_{t \to \infty} \frac {1}{2}(ln(t) -0) = \infty$$

The integral diverges. According to the integral test, the sequence also diverges.

Answer. The sequence diverges.