Question

1+1/a+1/a2+1/a3+.........upto n terms..find sum of the following series

Answer 1

Answer:

In case you're wondering where I got this question... First of all, I'm designing an RPG game where your meters upgrade based on your level. I also know the classic assymptotic increase towards 2:

1 + 1/2 + 1/4 + 1/8 ... = 2

I.e., 1/ai, i goes from zero to infinity, and the sum should equal 2. In this case, a equals 2. I forget what this series is called but I know it's a classical one, dating to zeno's arrow paradox I believe.

But then I thought, hmm, wouldn't it be interesting if I let your meters approach 3x their original value instead of 2? What would the series be then?

I actually found two ways to do it based on the classical one.

In the classical sum to 2, if we truncate the series and look at the values, we see 1 , 3/2 , 7/4 , 15/8, etc, all of which are closer and closer to 2 as it should be. The pattern here is (2i+1−1)/2i. It was easy to extrapolate this to a sum to 3: 1 , 8/3 , 26/9 , 80/27, etc, all of which are closer and closer to 3 as they should be. The pattern there is (3i+1−1)/3i.

However, to my eye, that pattern jumps a little too quickly towards 3. The jump from 1 to 8/3 = 2.667 is pretty big, so I rejected it.

The second way I found was to just go closer to 3 by 50% every time. So 1, 2, 2.5, 2.75, 2.875, etc. This doesn't "jump" like the last time, is still assymptotically approaching the value I want, and is an infinite series. However, it appears to use the exact same scheme as the classical sum to 2, just adding 1 to every term. So I consider it the same series as 1 + 1/2 + 1/4 ... but just offset by a constant. I guess you could say I aesthetically don't accept it.

(In fact, I think you can choose any value greater than or equal to 50% by which you go closer to your goal. E.g., 75% would be 1, 2.5, 2.875, 2.96875 etc. I think if it's less than 50%, the series won't converge, but this still leaves a lot of choices for this scheme.)

Then I thought of the third way, 1+1/a+1/a2+1/a3 ... = 3. But I don't know what a should equal.