1 last infinite series, power series

In summary: Hint: The total fraction of particles that escape at each side will be the fraction that impact there...
  • #1
Liquidxlax
322
0

Homework Statement



suppose a large number of particles are bouncing back and forth between x=0 and x=1, except that at each endpoint some escape. Let r be the fraction of particles reflected, so then you can assume (1-r) is the number of particles that escape at each wall. Suppose particles start at x=0 and head towards x=1; eventually all particles escape. Write and infinite series for the fraction at which escape at x=1 and x=0. Sum both series. What is the largest fraction of the particles which can escape at x=o



Homework Equations



sn-rsn = a(1-r^n)/(1-r)

0<r<1

The Attempt at a Solution




x=1 (1-r) + (1-r)^2... (1-r)^n

and same for x=0

sum

2(1-r) + 2(1-r)^2... + 2(1-r)^n
 
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  • #2
Liquidxlax said:
x=1 (1-r) + (1-r)^2... (1-r)^n

and same for x=0

You need to thinkl this through more carefully... the first impact is at x=1, what fraction of the original number of particles escapes? what fraction of the original number of particles doesn't? The next impact is at x=0 (not x=1) the first impact is at x=1, what fraction of the original number of particles escapes? what fraction of the original number of particles doesn't? ...and so on
 
  • #3
gabbagabbahey said:
You need to thinkl this through more carefully... the first impact is at x=1, what fraction of the original number of particles escapes? what fraction of the original number of particles doesn't? The next impact is at x=0 (not x=1) the first impact is at x=1, what fraction of the original number of particles escapes? what fraction of the original number of particles doesn't? ...and so on

at x=1 (1-r) escape and r is reflected... at x=0 (1-2r) escape and r/2 reflected? honestly this is supposed to be the easiest stuff, yet i have no problem with the apparent hard stuff...
 
  • #4
Liquidxlax said:
at x=1 (1-r) escape and r is reflected... at x=0 (1-2r) escape and r/2 reflected?

Not quite...the point is that the first impact with x=0 happens after the first impact with x=1. If there is a fraction [itex]r[/itex] left after the 1st impact with x=1 and a fraction (1-r) of those espaces, doesn't that mean that [itex]r(1-r)[/itex] escape the 1st impact at x=0, and [itex]r^2[/itex] are reflected?
 
  • #5
gabbagabbahey said:
Not quite...the point is that the first impact with x=0 happens after the first impact with x=1. If there is a fraction [itex]r[/itex] left after the 1st impact with x=1 and a fraction (1-r) of those espaces, doesn't that mean that [itex]r(1-r)[/itex] escape the 1st impact at x=0, and [itex]r^2[/itex] are reflected?

i did write that initially on paper, but i couldn't see it working. but i guess i could try again

so r^n(1-r) or (r^n - r^(n+1))
 
  • #6
Liquidxlax said:
i did write that initially on paper, but i couldn't see it working. but i guess i could try again

so r^n(1-r) or (r^n - r^(n+1))

I suggest making a table for the first few impacts at x=0 and at x=1, with the fraction (of the intial number of particles) that is reflected and the fraction that escapes.
 
  • #7
gabbagabbahey said:
I suggest making a table for the first few impacts at x=0 and at x=1, with the fraction (of the intial number of particles) that is reflected and the fraction that escapes.

is r changing with each term, or is it a constant fraction?
 
  • #8
Liquidxlax said:
is r changing with each term, or is it a constant fraction?


From the way the question is written, it is constant
 
  • #9
gabbagabbahey said:
From the way the question is written, it is constant

if that is true then

x=1 would be (1-r) + r^2(r-1) +... r^2n(r-1)

x=0 would be r(1-r) + r^3(r-1) +... r^(2n+1)(r-1)

so added together you get

(1-r) + r(1-r) + r^2(1-r) +... r^n(1-r)

Sn = ((1-r)(1-r^n))/(1+r)

well not sure about ^^^ because someone said that the max electrons leaving at x=1 as n approaches infinity is 1/2
 
  • #10
can anyone help me real quick, i just want to get my assignment done
 
  • #11
Liquidxlax said:
if that is true then

x=1 would be (1-r) + r^2(r-1) +... r^2n(r-1)

x=0 would be r(1-r) + r^3(r-1) +... r^(2n+1)(r-1)

so added together you get

(1-r) + r(1-r) + r^2(1-r) +... r^n(1-r)

Sn = ((1-r)(1-r^n))/(1+r)

well not sure about ^^^ because someone said that the max electrons leaving at x=1 as n approaches infinity is 1/2
That looks more like the number that are reflected...I thought you were supposed to find the fraction that escape
 
  • #12
gabbagabbahey said:
That looks more like the number that are reflected...I thought you were supposed to find the fraction that escape

aww crap, that is what i thought...


but what i thought i was doing initially was, taking the reflected particles and subtracting another fraction of what escapes by finding a new reflected amount say r^2 are reflected from r
 
  • #13
Liquidxlax said:
aww crap, that is what i thought...


but what i thought i was doing initially was, taking the reflected particles and subtracting another fraction of what escapes by finding a new reflected amount say r^2 are reflected from r

Hint: The total fraction of particles that escape at each side will be the fraction that impact there initially minus the total fraction that are reflected off that side
 
  • #14
gabbagabbahey said:
Hint: The total fraction of particles that escape at each side will be the fraction that impact there initially minus the total fraction that are reflected off that side

well w.e it is to late anyway, i had to hand in an incomplete assignment...
 

FAQ: 1 last infinite series, power series

What is an infinite series?

An infinite series is a sum of an infinite number of terms. It is represented in the form of "a + ar + ar^2 + ar^3 + ...", where a is the first term and r is the common ratio between each term.

What is a power series?

A power series is a specific type of infinite series where the terms are in the form of "c0 + c1x + c2x^2 + c3x^3 + ...", where c0, c1, c2, etc. are coefficients and x is a variable. It can be used to represent a function as an infinite polynomial.

What is the difference between an infinite series and a power series?

The main difference is that a power series includes a variable, while an infinite series does not. This means that the terms in a power series can change as the variable changes, while the terms in an infinite series remain constant.

How do you determine if an infinite series converges or diverges?

To determine convergence or divergence of an infinite series, you can use tests such as the ratio test, the root test, or the integral test. These tests compare the series to known convergent or divergent series to determine its behavior.

Can an infinite series have a finite sum?

Yes, an infinite series can have a finite sum if it converges. This means that as the number of terms in the series approaches infinity, the sum of all the terms remains finite. For example, the series "1/2 + 1/4 + 1/8 + ..." approaches a finite sum of 1 as the number of terms increases.

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