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Destroxia
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Homework Statement
How does a geometric series have a sum, or converge?
Homework Equations
Sum of Geometric Series = ##\frac {a} {1-r}##
If r ≥ ±1, the series diverges. If -1 < r < 1, the series converges.
The Attempt at a Solution
How exactly does a infinite geometric series have a sum, or converge (tend to) a specific limit?
I understand that it is due to partial sums that we are able to derive the formula for the sum of a geometric series, yet at the same time I don't understand how a sequence that will be always multiplied by itself to infinity can ever STOP and have a final sum, or how it converges (tends toward) a specific limit.
For example ##\sum\limits_{k=1}^{∞} (\frac {2} {3})^k##. This sum works out to 3, and does converge as r > -1 and r < 1. Why? The partial sums proof makes no sense to me, and how on Earth do things ever cancel to end up summing to 3?
Please explain to me if I have some conceptual misunderstanding of converges or sums of series, or if I'm just overlooking something...