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.
In: 248
> 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
The sum can only reach a finite value if your terms get increasingly smaller. If they didn’t, you’ll just keep adding stuff until you overshoot your estimated limit.
If you say *”the limit of infinitely adding 1s is 10″*, when you reach 8, you still have to add stuff (1, then 1, then 1) that is going to overshoot your target.
So a sum of infinite terms can only be finite if its terms trend towards zero.
Additionally, that condition is only necessary, not sufficient. `1/1 + 1/2 + 1/3 + 1/4 …` has terms that trend towards zero (for each ε > 0, you can find a natural number A so that 1/A < ε), but the infinite sum is not finite (for each L > 0, you can find a natural number N sot that SUM(for i from 1 to N of 1/i) > L)
* E.g. for ε = 0.00002, for any natural A >= 50000 you have 1/A < ε
* E.g. [for L = 10, for any natural N >= 30000](https://www.wolframalpha.com/input?i=sum+of+1%2Fn+for+n+in+%281..30000%29) you have SUM(for i from 1 to N of 1/i) > L
Latest Answers