# Workshop: Session 6 — Summary

We began by looking at some of the lovely mathematics of infinite series. Working with the **geometric series**

we explored **partial sums**, the formula for the sum of an **infinite geometric series**, and some ideas in **Calculus**.

By looking at the **partial sums **of this series, a lot of good ideas surface:

You can quickly generalize to the formula

With this, you can predict the answer (**1!**) and discuss ideas like limits.

Still working with the above series, you can explore the way in which we derive the formula for the **sum of an infinite gepmetric series**.

Take the original series, set it equal to **S**

**(Eq 1)**

multiply this equation by **2**

to get

**(Eq 2)**

Then subtract **Eq1** from **Eq2** to get the magical result

A nice geometric interpretation of this can be seen in this Proof Without Words.

The same approach produces the general formula for the sum of infinite geometric series:

We also had some fun looking at series where **r**, the common ratio**, **didn’t satisfy the requirement .

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