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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So this is a difficult question but here I go:
– Let us say we look at the infinite sum where each term being added is 1:
1+1+1+1…=x
We can easily see that no matter how many 1s you add, the number will just keep on growing and growing. We call this “divergent”
– Let us say we look at the infinite sum where each term being added is a tenth of the previous term:
1+0.1+0.01+0.001…=x
We can easily see that no matter how far you go down the line you will only ever get to the number 1.11111111… . We call the fact that some infinite sums settle at a particular value “convergent”
It is not always easy to find out if a sum is convergent or divergent, but we have tricks that help us figure it out.
It has nothing to do with all terms being less than 1. For example the sum of an infinite 1/2 is divergent. 1/2+1/2+1/2+1/2=x. This keeps growing and growing at half the rate of 1+1+1+1…, but the 1/2 sum will reach all numbers that the 1 sum will reach.
Similarly, the inverses of the integers: 1+1/2+1/3+1/4+1/5+1/6… Is well known to diverge.
How do we know this:
Look at this sum:
– 1+1/2+1/4+1/4+1/8+1/8+1/8+1/8+1/16….
At each point in the sum, that sum is smaller than this sum:
– 1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9…
However, notice that if we group the smaller sum like this and add the terms in the brackets you get: 1+(1/2)+(1/4+1/4)+(1/8+1/8+1/8+1/8)+(1/16….
1+1/2+1/2+1/2+… (we have already shown that An infinite sum of 1/2 diverges, so the other larger sum must also diverge)
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