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## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>Unit 10

Lesson 2: Working with geometric series

# Proof of infinite geometric series formula

AP.CALC:
LIM‑7 (EU)
,
LIM‑7.A (LO)
,
LIM‑7.A.3 (EK)
,
LIM‑7.A.4 (EK)
Say we have an infinite geometric series whose first term is a and common ratio is r. If r is between minus, 1 and 1 (i.e. vertical bar, r, vertical bar, is less than, 1), then the series converges into the following finite value:
limit, start subscript, n, \to, infinity, end subscript, sum, start subscript, i, equals, 0, end subscript, start superscript, n, end superscript, a, dot, r, start superscript, i, end superscript, equals, start fraction, a, divided by, 1, minus, r, end fraction
The AP Calculus course doesn't require knowing the proof of this fact, but we believe that as long as a proof is accessible, there's always something to learn from it. In general, it's always good to require some kind of proof or justification for the theorems you learn.

## First, let's get some intuition for why this is true. This isn't a formal proof but it's quite insightful.

Infinite geometric series formula intuitionSee video transcript

## Now we can prove the formula more formally.

Proof of infinite geometric series as a limitSee video transcript

## Want to join the conversation?

• where did you get the -ar^(n+1), that's not in first video
• why |r| must be lower than 1?
mathematically its correct that we reach to some specific value for S sub infinity value. But it is negative.
I know that for r=1 this formula cant help us.
• If |r|≥1, then the size of the terms doesn't decrease. So for any value you might think this converges to, it eventually exceeds it.
• Why is the second proof more formal than the first? Is it the approach? Starting from a finite sum and taking an infinite limit?
• The first proof was less formal because we assumed that the sum converged. That's a necessary thing to assume/prove if we're going to treat S like any other real number that can be moved around the equation.

The second proof was more rigorous because we didn't assume |r|<1, and explored what would happen to the sum if r took on different values.