They both get closer to but never reaching zero. Is the reason simply that one gets 1/x^2 gets closer to zero faster? So whats the threshold for something to be considered finite or infinite?
In: 59
The problem with 1/x is as follows:
You start with 1.
Now to get another 1/2, how many more terms do we need?
In this case just 1 more term
Now to get another 1/2 how many do we need?
Next two are 1/3 and 1/4. Both of these are greater than 1/4. So 1/3 +1/4 >=1/2.
So 2 terms
Okay, how many for the next 1/2?
Similarly the next 4 are 1/5, 1/6, 1/7, 1/8
1/5 + 1/6 + 1:7 + 1/8 >= 4 * 1/8 >= 1/2.
So 4 terms
Etc
So basically, you can keep adding 1/2 forever, you just need double the number of terms each time. But you have infinite terms, so you can reach any arbitrary sum you wish
This isn’t true for 1/x^2. Feel free to try it and see why
Edit: formatting.
Edit: note that the area under the curve is equivalent to the sum of the series for the purposes of this. Slightly different problem but this is easier to explain
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