## Learn more physics!

*Most recent answer: 06/28/2017*

Q:

What is(1 - 1/10)(1 - 1/11)(1 - 1/12)...(1 - 1/99)(1 - 1/100)?

- Bailey (age 12)

USA

A:

You can do that exact calculation on any calculator. Here's how I'd make a guess.

The log of a product is the sum of the logs.

ln(1-1/x) is about -1/x for x>10. So the log of that product is close to the integral from 10 to 100 of -1/x, which is just ln(10/100)= ln(1/10). So the ln of your product is just ln(1/10) so your product is about 1/10.

Let us know how close that is.

p.s.- My argument above was ok as far is it went but way too complicated and not accurate enough. Try looking at just the first few terms: (9/10)(10/11)... You'll see an extremely simple pattern, without any of the fancy math I used above. It will give you the exact answer!

Mike W.

*(published on 06/28/2017)*

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