Question 1194656: For the infinite geometric series below, identify whether it converges or diverges.
10, 5, 2.5, 1.25, ....
Select one:
a.
Diverges
b.
Both
c.
Neither
d.
Converges
Found 2 solutions by greenestamps, math_tutor2020:
Answer by greenestamps(13198)   (Show Source):
You can put this solution on YOUR website!
The common ratio is 5/10 = 1/2.
An infinite geometric series converges if (and only if) the ratio r satisfies -1 < r < 1.
Is 1/2 between -1 and 1?
Answer by math_tutor2020(3816)  (Show Source):
You can put this solution on YOUR website!
Answer: D) converges
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How to find that answer:
We'll need the common ratio, denoted as r.
Pick any term and divide it over the previous term
r = term2/term1 = 5/10 = 0.5
r = term3/term2 = 2.5/5 = 0.5
r = term4/term3 = 1.25/2.5 = 0.5
We get the same common ratio (r = 1/2 = 0.5) each time, which helps confirm that we do indeed have a geometric sequence.
If r differed at all in any of the results above, then the sequence of course wouldn't be geometric.
Notice that -1 < r < 1 is the case.
r = 1/2 = 0.5 is between -1 and 1
This is sufficient criteria to confirm that the infinite geometric series converges to some value.
Having r in this range means we are adding smaller pieces to the partial sums to slowly approach some finite value.
Consider the case that r > 1. Such that r = 2 for instance. It should be fairly obvious the geometric sequence terms grow forever out of control. There's no way to approach a finite value.
Anyways, because r = 1/2 = 0.5, this means -1 < r < 1 is true. Therefore, the infinite geometric series converges
We can stop here since we've concluded the answer is choice D
This of course rules out choice A) diverges because that is the complete opposite of converging.
Choice B is ruled out as well. It is impossible for a geometric series to BOTH converge and diverge.
That would be like saying a door is fully closed and completely open at the same time.
This allows us to rule out choice B. Choice C is a similar story.
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Extra stuff to consider. This section is optional.
If you are curious what the series converges to, then,
a = first term = 10
r = common ratio = 1/2 = 0.5
S = sum of the infinitely many geometric terms
S = a/(1-r)
S = 10/(1-0.5)
S = 10/0.5
S = 20
So if you were to add 10+5+2.5+1.25+... and go on forever doing this, then you'll land on 20.
Partial sums:- S2 = 10+5 = 15 = sum of the first two terms
- S3 = 10+5+2.5 = 17.5 = sum of the first three terms
- S4 = 10+5+2.5+1.25 = 18.75 = sum of the first four terms
- S5 = 10+5+2.5+1.25+0.625 = 19.375 = sum of the first five terms
and so on...
Even when we get to something like S100, the sum of the first 100 terms, we'll never actually hit 20 itself. We'll get very close to it however.
As you can probably guess, there's a nice recursive nature going on.
That relationship is
Two examples:
S3 = S2 + a3
S3 = 15 + 2.5
S3 = 17.5
and
S4 = S3 + a4
S4 = 18.75 + 0.625
S4 = 19.375
This trick allows us to compute S5, S6, ... without having to write out long strings of summations.
Simply add on the newest term  to the previous partial sum.
The drawback though is that if we wanted to compute S100 this way, we'd have to know what S99 was. Then in turn we'd have to know S98, and S97, and so on.
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