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 [itex]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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