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

  • Thread starter DivineNathicana
  • Start date
In summary, the following simplified fraction is obtained by multiplying the fractions inside of parentheses. It is 1-1/2, 1-1/3, 1-1/4, 1-1/5... up to (1-1/n). If n=the denominator of the first of the two fractions being multiplied, then the result is infinity.
  • #1
DivineNathicana
57
0
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
 
Physics news on Phys.org
  • #2
Add the fractions inside each set of parentheses and see if there is a pattern! :-)
 
  • #3
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...
 
  • #4
[tex]\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}[/tex]

Do you see a pattern yet?
 
  • #5
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.
 
  • #6
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

[tex]\frac {n-1}{n} = 1 - \frac {1}{n}[/tex]

Now let n go to infinity! :-)
 
  • #7
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?
 
  • #8
DivineNathicana said:
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?
 
  • #9
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?
 
  • #10
Exactly! I knew you'd see it sooner or later. :-)
 
  • #11
Haha thank you! Maybe next time I should try to get started a bit earlier...
 

1. What is a "simplifying (1-1/n) sequence" in terms of simple math?

A "simplifying (1-1/n) sequence" is a mathematical sequence that follows the pattern of subtracting 1 from the previous term, then dividing by n, where n is a positive integer. This sequence can be written as 1, (1-1/n), (1-1/n)^2, (1-1/n)^3, and so on.

2. How do you find the limit of a simplifying (1-1/n) sequence?

The limit of a simplifying (1-1/n) sequence as n approaches infinity is 1. This means that as n gets larger and larger, the terms in the sequence get closer and closer to 1.

3. Can a simplifying (1-1/n) sequence have negative terms?

No, a simplifying (1-1/n) sequence cannot have negative terms because the pattern always starts with 1 and then subtracts 1 from the previous term, resulting in only positive terms.

4. What is the formula for finding the nth term in a simplifying (1-1/n) sequence?

The formula for finding the nth term in a simplifying (1-1/n) sequence is (1-1/n)^(n-1). This can also be written as 1/n^(n-1).

5. How can simplifying (1-1/n) sequences be used in real-life applications?

Simplifying (1-1/n) sequences can be used in various real-life applications such as in population growth or decay, compound interest calculations, and in calculating the probability of events occurring. They can also be used in computer algorithms and in the study of fractals.

Similar threads

  • Introductory Physics Homework Help
Replies
5
Views
750
  • Introductory Physics Homework Help
Replies
1
Views
148
  • Introductory Physics Homework Help
Replies
1
Views
859
  • Introductory Physics Homework Help
Replies
3
Views
847
  • Introductory Physics Homework Help
Replies
11
Views
1K
Replies
1
Views
299
  • Introductory Physics Homework Help
Replies
5
Views
686
  • Introductory Physics Homework Help
Replies
8
Views
503
  • Introductory Physics Homework Help
Replies
12
Views
684
  • Introductory Physics Homework Help
Replies
3
Views
818
Back
Top