Summing Infinite Series

Why Bother with Infinite Series

After all, aren't they difficult?

Why is the Subject of Infinite Series Difficult?

It is not intuitive.

What is the sum of an infinite series?

It isn't so obvious.

Convergent Infinite Series

The notion of sum as limit of the sequence of partial sums.

Is the Order of Terms Important?

There is a big difference between absolutely convergent series and conditionally convergent series.

Summable Infinite Series

What about the sum of the powers of i (the square root of -1)?

The notion of distance

There is more than one kind of distance. Each kind of distance determines a kind of limit.

The uniform metric

Distance between functions on an interval rather than distance between numbers.

Is there any way to sum the series 1 + 2 + 2² + 2³ + ... ?

Well, maybe. It's all a question of what the rules are.

When the answer is no:

• The series of powers of 2 is not convergent.
• The series of powers of 2 is not Cesaro summable.

When the answer is maybe yes:

• The notion of analytic continuation.
  If it's sometimes true, why can't it be always true.
  
  
• The subject of 2-adic analysis.
  Stepping away from the real numbers (but staying in the real world)
  
  
• The subject of z-adic analysis.
  How to imagine every power series as convergent