Select Page

Question:

True/False?

The series $$1-\frac{1}{\sqrt2}+\frac{1}{\sqrt3}-\frac{1}{\sqrt4}+…$$ is divergent.

Hint:

Recall the alternating series test (or the Leibniz test)

Discussion:

Let $$a_n=\frac{1}{\sqrt{n}}$$. The alternating series test says that if we have a series like $$a_1-a_2+a_3-a_4+…$$ then a sufficient condition for the convergence of this series is: $$a_n$$ is decreasing and $$a_n\to 0$$ as $$n\to \infty$$.

Here, $$a_n$$ satisfies the above condition.

Therefore, the series converges.