Simple Math: Simplifying (1-1/n) Sequences

[DivineNathicana]

Greetings. Alright, if anyone's bored enough to be on-line right now, what is the following simplified and how do you get it?

(1-1/2)(1-1/3)(1-1/4)(1-1/5)...(1-1/n)

Thanks for any help,

- Alisa

 

[Tide]

Add the fractions inside each set of parentheses and see if there is a pattern! :-)

 

[DivineNathicana]

I got infinity(1-2/n+2), where n= the denominator of the first of the two fractions being multiplied. That doesn't sound too solid...

 

[Tide]

$$\left(1 - \frac {1}{2}\right) \left(1 - \frac {1}{3}\right) \left(1 - \frac {1}{4}\right) \cdot \cdot \cdot \left(1 - \frac {1}{n}\right) = \frac {1}{2} \cdot \frac {2}{3} \cdot \frac {3}{4} \cdot \cdot \cdot \frac {n-1}{n}$$

Do you see a pattern yet?

 

[DivineNathicana]

I see the pattern, but I still keep on getting weird-looking answers such as

∞!
---------
((∞-1)!+1)

The (----) being a division sign. If the symbol doesn't come out, it's supposed to be infinity.

 

[Tide]

Well, first off, your original post said nothing about extending it to infinity. But since that seems to be where you are headed consider that

$$\frac {n-1}{n} = 1 - \frac {1}{n}$$

Now let n go to infinity! :-)

 

[DivineNathicana]

Wait up, it's 2 A.M., and I can't think very straight. Why does (n-1)/n=1-(1/n)? And shouldn't we be doing factorials like ((n-1)!)/n! or something like that since all of this has to be multiplied?

 

[Tide]

Wait up, it's 2 A.M., and I can't think very straight. Why does (n-1)/n=1-(1/n)? And shouldn't we be doing factorials like ((n-1)!)/n! or something like that since all of this has to be multiplied?

Um ... it's a fundamental property of numbers? The distributive property.

You can certainly use factorials but why would you want to when all the intermediate factors cancel out?

 

[DivineNathicana]

Ooh sorry haha I didn't realize what you were talking about. Okay, yeah, so (n-1)/n= 1-(1/n), I see that. So then wouldn't it be just 1/n if we consider all the factoring out?

 

[Tide]

Exactly! I knew you'd see it sooner or later. :-)

 

[DivineNathicana]

Haha thank you! Maybe next time I should try to get started a bit earlier...