1 last infinite series, power series

[Liquidxlax]

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

 

[gabbagabbahey]

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

 

[Liquidxlax]

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...

 

[gabbagabbahey]

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?

 

[Liquidxlax]

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))

 

[gabbagabbahey]

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.

 

[Liquidxlax]

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?

 

[gabbagabbahey]

is r changing with each term, or is it a constant fraction?

From the way the question is written, it is constant

 

[Liquidxlax]

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

 

[Liquidxlax]

can anyone help me real quick, i just want to get my assignment done

 

[gabbagabbahey]

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

 

[Liquidxlax]

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

 

[gabbagabbahey]

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

 

[Liquidxlax]

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...