Question

Give one example for the complex series which is

1) Conditionally Convergent.

2) Absolutely Convergent.

3) Both.

4) Neither.

Answer 1

1] For conditional convergence;

$\displaystyle \sum^\infty_{n=1}\frac{(-1)^n}{n}$

2] For absolute convergence;

$\displaystyle \sum^\infty_{n=1}\frac{i^n}{n^2}$

3] Both absolutely and conditionally convergent; Does not exist since from definition, a complex series that converges but is not absolutely convergent is said to be conditionally convergent. Thus they are exact opposites.

4] For neither;

$\displaystyle \sum^\infty_{n=0}(1-i)^n$

is a divergent series.