- 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 + 2^2 + 2^3 + ... ?
- 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

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