$\sum_{n=0}^{\infty} \frac{(-1)^n}{1+\sqrt n}$

Question

$\sum_{n=0}^{\infty} \frac{(-1)^n}{1+\sqrt n}$.

Find out whether this series is absolutely convergent or conditional convergent or divergent.

Now integral test gives me that this series is not absolutely convergent and the term goes to zero so this is conditional convergent.

Am I right in this case?

Answer 1

Yes, you are right.

In order to prove that it is not absolutely convergente, I would have used the comparison test, comparing $\frac1{1+\sqrt n}$ with $\frac1{\sqrt n}$, but that's a matter of taste.

Answer 2

HINT

Note that

• for absolutely convergence consider limit comparison test with $\sum \frac1{\sqrt n}$

• for conditional convergence note that $\frac1{1+\sqrt n}\to 0$ is strictly decreasing

Answer 3

Just to stress a point made by gimusi, to prove conditional convergence of an alternating series $\sum(-1)^na_n$ (with $a_n\gt0$ for all $n$), it's not enough to show that $a_n\to0$. You need to show that $a_n$ decreases to $0$. A useful counterexample to keep in mind is

$$1-{1\over2^2}+{1\over3}-{1\over4^2}+{1\over5}-{1\over6^2}+\cdots$$

which diverges even though the terms are clearly tending to $0$.