Answer to Question #295143 in Complex Analysis for Jofar

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.