Oh wow, a math question came up at work Finite series

In summary, a aerospace engineer was trying to solve a math problem that he had difficulty with, and found that he could rewrite it as a geometric series. He credits his degree in physics for helping him to solve the problem.
  • #1
blochwave
288
0

Homework Statement


Being professionals now we've all forgotten our math skills and I'm trying to impress everyone. P=sum from k=0 to n of (x)^(m-k)*(1-x)


Sorry for the hurried lack of latex, it's x^(m-k), and that term is multiplied by (1-x)


Homework Equations


Uh-oh


The Attempt at a Solution



Unfortunately this isn't really homework and I don't even know where to start, so I guess it'd be too much to ask to just do it! If someone could just get me started methinks hazy memories could kick in
 
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  • #2
Hi blochwave! :smile:

Well, most of it is constant, so it's (x^m)(1-x)∑x^(-k),

(or is it (x^-m)(1-x)∑x^k ?)

so all you have to sum is ∑x^(-k). :smile:

(or ∑x^k)
 
  • #3
I assume you mean,

[tex]P = \sum_{k=0}^{n} x^{n-k}(1-x) = (1-x)\sum_{k=0}^{n}\frac{x^n}{x^k}[/tex]

[tex] = (1-x)\left(x^n+x^{n-1} + x^{n-2} + \ldots + x^2 + x +1\right)[/tex]

[tex] = \left(x^n+x^{n-1} + x^{n-2} + \ldots + x^2 + x +1\right) - \left(x^{n+1}+x^{n} + x^{n-1} + \ldots + x^3 + x^2 +x\right)[/tex]

Notice that all the terms where the exponent is between n and 1 inclusive cancel leaving,

[tex]P = 1-x^{n+1}[/tex]

Edit: Ooops, I thought the m was an n, never mind. See TT's post.
 
  • #4
You guys rock so hard

EDIT: Unfortunately I don't

A)It WAS x^(m-k), m is a constant distinct from n, sorry

it was (1-x)^k, to make it I believe more difficult
 
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  • #5
blochwave said:
You guys rock so hard

EDIT: Unfortunately I don't

A)It WAS x^(m-k), m is a constant distinct from n, sorry

it was (1-x)^k, to make it I believe more difficult
So the series is,

[tex]P = \sum_{k=0}^{n}x^{m-k}\left(1-x\right)^k[/tex]

Correct?
 
  • #6
Yes, I wrote it down this time to avoid further embarrassment >_>
 
  • #7
The contribution from the x(m-k) term will be similar to the series detailed in post #3. For the contribution of the parenthesised term, one may consider using the binomial theorem.

Edit: I'm curious as to your line of work, in what context did the series arise?
 
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  • #8
I don't even know what he's doing it for, it's something to do with probabilities like I said

Check this though: I did the ratio convergence test to make sure he didn't ask a stupid question, forgetting that it only works if the series is geometric

Well if you divide any subsequent terms you get x^-1(1-x), which I realized is r

so the series can be written as x^m[r]^k, if a=x^m, r is that thing above(I checked this for at least the first few terms), you can just write down the solution to a geometric series using those terms and BAM

Did I do good? He's an aerospace engineer and I have a degree in physics, we work for an engineering consulting type place. I felt obliged to solve this after I didn't know his other question of which was more efficient, a turbofan or turboprop :(

Edit: Well duh, it's a finite series, I shouldn't have even been doing a convergence test, BUT in the process I found out I could rewrite it as a geometric series, and right is right. I hope
 
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  • #9
You know what, that's real nice blochwave, I didn't spot that. Looks good to me :approve:
 

1. What is a finite series in math?

A finite series in math is a sequence of numbers that has a specific starting point and a specific ending point. It is a sum of a finite number of terms and can be represented using sigma notation.

2. How do you find the sum of a finite series?

To find the sum of a finite series, you can use the formula S = (n/2)(a + l), where n is the number of terms, a is the first term, and l is the last term. Alternatively, you can use the sigma notation formula, where the lower limit is the starting point and the upper limit is the ending point.

3. What is the difference between a finite and infinite series?

A finite series has a limited number of terms, while an infinite series has an endless number of terms. In other words, a finite series has a specific starting and ending point, while an infinite series continues infinitely.

4. Can a finite series have negative terms?

Yes, a finite series can have negative terms. The terms in a finite series can be either positive or negative, depending on the sequence of numbers.

5. How is a finite series used in real life?

A finite series is used in many real-life situations, such as calculating loan payments, estimating population growth, and calculating the total cost of items in a shopping list. It is also used in fields like engineering, finance, and physics to solve real-world problems.

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