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Proof of infinite geometric series formula

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.

Khan Academy video wrapper
Infinite geometric series formula intuitionSee video transcript

Now we can prove the formula more formally.

Khan Academy video wrapper
Proof of infinite geometric series as a limitSee video transcript

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