does this apply only for terms that are less than 1 eg 1 + 1/2 + 1/4…. or does this apply to all ap/gp. I remember studying this, but it’s been so long I remember only the gist.
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Lots of good answers for the S1 = 1 + 1/2 + 1/4 + … series.
But I’ll offer a general answer:
– under what conditions does a series converge on a finite value?
There are several “tests” we use.
One is looking at the Partial Sum – does the total sum of the series get closer and closer to a value without any back-stepping? S1 = 1 + 1/2 + 1/4 + … does this, as been demonstrated by others here.
But what about S2 = 1 – 1/2 + 1/3 – 1/4 + 1/5 – … and so on? It goes back and forth. Does S2 converge? This osculating partial sum needs to be treated differently.
And what about S3 = 1 + 1/2 + 1/3 + 1/4 + … does it converge? Not this one! This one is quite special – we call it the https://en.wikipedia.org/wiki/Harmonic_series_(mathematics) We know it doesn’t because we can see the sum of the first 1, 2, 4, 8, … terms each of themselves never goes below 1.
Now what about S4 = 1 – 1 + 1 – 1 + 1 … ? Well, it never blow up to infinity, but it also never “settles” on 1 value. So no.
There is a deep pool of math exploring which series converge and which do not. And then those that do not, what do do with them.
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