Quote by AlephZero
The fact that you are asking this question when you have ''learned about infinite geometric progressions'' is good, because it shows you are interested in math and thinking about what you are doing, and not just learning how to pass the next test.
There are lots of areas of math where "simple" questions turn out to be hard to answer. The sum is actually ##\log_e 2## = approximately 0.6931, but you need to learn calculus to understand why that is the answer.
If you want to try to find the sum by hand (or with a computer), then as post #2 said, you have to add the numbers up in the same order as the original series. If you rearrange them, you can get more or less any answer you like.
You can get two estimates that are too big and too small by taking the terms in pairs.
1  1/2 = 1/(1.2)
1/3  1/4 = 1/(3.4)
1/5  1/6 = 1/(5.6)
etc
So the sum = 1/2 + 1/12 + 1/30 + ...
Or, take the first term on its own and the rest in pairs.
1/2 + 1/3 = 1/(2.3)
1/4 + 1/5 = 1/(4.5)
1/6 + 1/7 = 1/(6.7)
etc
So the sum = 1  1/6  1/20  1/42 ....
That will give you two estimates that bracket the answer, but this series is very slow to converge, so you will have to take hundreds of terms to find the answer to a few decimal places.

How you got the answer ln2. that's exactly the answer of this question.
adding them like simple sum is dummy style.
getting sum of this series is part of question in physics. So i must have been read that in mathematics but not getting that.
this is part of an electrostatics question where i am required to find potential at origin when negative charges are placed at odd positions (on x axis) and positive charges are placed at even position.
You say i need to know calculus for it. i think i know. Please see my
syllabus(calculus is at lower portion) and give me a proper hint to approach this question.