## Coursera Complex Analysis 6.1 Infinite Series

### Aug 8, 2016

## Slide 2 Definition

which converges if *sequence of partial sums*, Sn, converges

## Slide 3 Example

Formula in usual way

Since as long as |z| < 1

then for |z| < 1.

## Slide 4 Convergence and Divergence

Theorem: If a series converges, then (*)

The converse (swap propositions), if then sum converges is **not** necessarily true – classic example is the harmonic series.

Contrapositive (*) then series **diverges**, of course is true.

For 1/(1 – z) , this diverges when |z| >= 1

## Slide 5 Real and Imaginary parts of series

z = so

So

<< more , separate real and imaginary parts>>

## Slide 6 Another example of convergence. Part I

Does converge?

If take modulus of the terms, then get harmonic series which does not converge.

What about original series on this slide? Split into real and imaginary parts.

## Slide 7 Another example of convergence. Part II

Preliminaries: k is even, ( a real number)

Similarly, k is odd, (a *purely* imaginary number)

So (note the limits).

Using prelim results and simplifying:

The first sum is the alternating harmonic series which converges (can justify by looking at intervals on real line)

and (probably) because denominators of terms second series getting smaller and series is alternating this also converges.

## Slide 8 Absolute Convergence

Definition ? converges absolutely if the series converges

<< examples >>

If converges absolutely then it converges and converges absolutely

## Slide 9 Example of absolute convergence inequality

<< to do >>