Multiply out these expressions:

(1 +r)(1 - r),\\ (1 +r + r^2)(1 - r),\\ (1 +r + r^2 + r^3)(1 - r),\\ (1 +r + r^2 + r^3 + r^4)(1 - r),\\ ...

What do you notice?

Does this pattern continue?

Can you prove it?

What does this tell you about the sum to n terms of the geometric series with first term 1 and common ratio r?

Why do we get a formula for all values of r except r = 1?

Explain the formula:

\sum_{i=0}^{n-1}r^i = 1 + r + r^2 + r^3 + ... +r^{n-1} = \frac {1-r^n}{1-r} for all r \neq 1.

What can you say about the limit of r^n for -1<r<1 as r\to \infty?

What does this suggest to you about the infinite sum of the geometric series:

for -1<r<1, \ \sum_{i=0}^{\infty} r^i = (1 +r + r^2 + r^3 + ... )?

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