What is the solution to this simple sum problem?

[mathal]

(1/1-1/6) + (1/2-1/7) + (1/3-1/8)...+(1/95-1/100)
This was on the cover of a local paper with the caption "You can't solve this but he can!" The 'he' in the caption was a 12 year old boy. On the next page they gave as his solution 2.2. Two things are clear,
1.-the boy had the right answer and
2.-the editor cut the answer to one decimal point for brevity.
I was wondering, while I drove home from where I saw the cover and second page, what is the actual point one should stop adding the bracketed values to get closest to the abbreviated answer in the paper of 2.2. It took me about a minute to solve it in my head while driving.
No pen, paper and DEFINITELY no calculator required. Be as brief as possible in your explanation of the simplest solution.
mathal

 

[CompuChip]

My initial thoughts:

My first observation is that the answer will be 1/1 + 1/2 + 1/3 + 1/4 + 1/5 - 1/96 - 1/97 - 1/98 - 1/99 - 1/100 as all the other values cancel out.

Using the first 5 terms rounded to 1 or 2 decimals you would get 1 + 0.5 + 0.33 + 0.25 + 0.2 = 2.28.

If you want to make it more precise you could subtract 5 * 1/100 = 0.05 to get 2.23.
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I checked this using a calculator and it is actually correct to the given precision.

But I'll be happy to hear a smarter approach.

 

[mathal]

My initial thoughts:

My first observation is that the answer will be 1/1 + 1/2 + 1/3 + 1/4 + 1/5 - 1/96 - 1/97 - 1/98 - 1/99 - 1/100 as all the other values cancel out.

Using the first 5 terms rounded to 1 or 2 decimals you would get 1 + 0.5 + 0.33 + 0.25 + 0.2 = 2.28.

If you want to make it more precise you could subtract 5 * 1/100 = 0.05 to get 2.23.
[/color][/size]
I checked this using a calculator and it is actually correct to the given precision.

But I'll be happy to hear a smarter approach

Spoiler

Your answer is to the original puzzle that the kid solved...-no problem... I saw that without a calculator and also knew it was only an approximation to the real value which is necessarily a repeating decimal fraction. The puzzle I am presenting is to determine the point in this series of additions that will result in the closest approximation to 2.2 in this series.The 2.2, I assume is the editor's truncation of an already truncated solution he/she called for to save space. The puzzle is dog simple, as I pointed out in the intro header. This is more of an AHA experience than anything else. Try again without a calculator.

mathal

.

This is a reply to the last post. Ignore it if you want to solve this without any help.

mathal

 

[mathal]

This post contains the answer inside the spoiler.
For a DOH! moment read it. I'd stick with the AHA moment, but to each his/her own.

Spoiler

1+1/2+1/3+1/4+1/5 -2.2 is the sum of the last 5 negative terms needed.
2+17/60 -(2 +12/60) =5/60
The last term to add to get closest to the abreviated answer of 2.2 is (1/57-1/62)

mathal

 

[mathal]

I'm going to leave this last message for clarification. CompuChip submitted a message in which he explained the math series in the newspaper that inspired this puzzle. His open comments indicate he felt he was submitting an answer to the puzzle. He was mistaken. The answer is in my last post. If you really need help in understanding what is involved in the math series in the newspaper article read his explanation. It is correct. The puzzle was explained in the opening post after mentioning where the idea came from.
mathal

Good puzzles are a gestalt.