Answer to Question #347915 in Real Analysis for Nikhil

Question

If (an) is convergent then $\sum_{i=1}^{\infty} a_n$ is also convergent. True or false with the full explanation.

Answer 1

The sequence $a_n=\dfrac{1}{n}$ converges to $0.$

But the harmonic series

$$\displaystyle\sum_{i=1}^{\infin}\dfrac{1}{n}$$

is divergent.

This is the counterexample of the statement.

Therefore the statement is False.