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Μάθημα: Ολοκληρωτικός Λογισμός > Ενότητα 5
Μάθημα 2: Infinite geometric series- Worked example: convergent geometric series
- Worked example: divergent geometric series
- Infinite geometric series
- Infinite geometric series word problem: bouncing ball
- Infinite geometric series word problem: repeating decimal
- Proof of infinite geometric series formula
- Convergent & divergent geometric series (with manipulation)
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Proof of infinite geometric series formula
Say we have an infinite geometric series whose first term is and common ratio is . If is between and (i.e. ), then the series converges into the following finite value:
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.
Now we can prove the formula more formally.
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