Question

Give one example and also prove that for the complex series which is

1) Conditionally Convergent.

2) Absolutely Convergent.

3) Both.

4) Neither

Answer 1

1. The series $\sum\limits_{n=1}^{+\infty}\frac{(-1)^n}{n}$ is conditionally convergent, because its terms are alternating, and their modulus tends to 0 (Leibniz' test for alternating series).

The series is not absolutely convergent, since the series of modules of terms is a harmonic series that diverges.

2. The series $\sum\limits_{n=1}^{+\infty}\frac{1}{n(n+1)}$ is absolutely convergent, because its terms are positive and the sequence of the partial sums converge to 1:

$S_N=\sum\limits_{n=1}^{N}\frac{1}{n(n+1)}=\sum\limits_{n=1}^{N}(\frac{1}{n}-\frac{1}{n+1})=$

$=(1-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+\dots+(\frac{1}{N}-\frac{1}{N+1})=1-\frac{1}{N+1}\to 1$

3. The example from part 2 is the series that is both absolutely and conditionally converges, because any absolutely convergent series is also simply (conditionally) convergent.

4. The series $\sum\limits_{n=1}^{+\infty}(-1)^n$ is neither absoilutely or conditionally convergent.

Really, since its terms do not tend to 0, the series cannot be divergent.