What is the relationship between power series and exponential functions?

[tigigi]

1. I this from my homework solution.

(1-t/s)^n = exp(-t/s)
as n goes to infinity

I don't understand. I checked the exponential power series. It should be :
exp(x) = summation (x^n / n!)
n=0 to infinity

How come it could be a exponential function ?

2. another is that why

<t> = integral from 0 to infinity (t*P(t) dt) ?
average t

P(t)dt = probability that an electron has no colission till time t *
probability that it has a collision between time t

probablitiy has no collision is exp(-t/s)

t+dt = exp(-t/s) *dt/s

Thanks a lot !

 

[D H]

1. This is obviously wrong:

\sum_{n=0}^{\infty}(1-t/s}^n \ne \exp(-t/s)

You don't need to expand the series. Use t/s=0.5. The series evaluates to 2, which is obviously not 1/\sqrt e.

I suspect you are missing something here.

2. In general, the expected value of some random variable X with respect to some probability density function p(t) is defined as \int X p(t)dt, where the integration is performed over the domain of the PDF. Here, you want the expected value of the time until a collision for an exponentially distributed collision time. BTW, the PDF is p(t)=\exp(-t/s)/s[/tex], not \exp(-t/s).

 

[tigigi]

oops, I should make the notation more clear.


the ans says that

( 1- dt/s )^n = exp ( -t/s ) as n goes to infinity

there's no summation here.

I'm wondering where this comes from ?

 

[D H]

This is still not right. This is:

\exp\left(-\;\frac t s\right) \equiv \lim_{n\to\infty} \left(1-\frac 1 n\;\frac t s\right)^n

 

[HallsofIvy]

It's not a matter of "more clear"- you didn't say anything about a limit before!

Oh, and now you have "dt/s" where before you had "t/s". Was that a typo?

Of course, for fixed t and s, 1- t/s is just a number. If that number is larger than 1, the limit is \infty, if that number is less than 1, the limit is 0, if that number is equal to 1, the limit is 1. It certainly is not "e^-t/s".

It is true that \lim_{n\rightarrow \infty} (1+ \frac{x}{n})^n= e^{x}
Replacing x by -t/s, we get
\lim_{n\rightarrow \infty} (1- \frac{x}{ns})^n= e^{-t/s}

Is it possible that you are missing an "n" in the denominator?

 

[tigigi]

I got it. Thanks a lot ! :)

 

[tigigi]

yes, I think so. There's an n missing in the denominator.

 

[absolute76]

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