Asymptotic Expansion Homework: I(x) Integral

[the_kid]

Homework Statement

Calculate an asymptotic expansion to all orders of I(x) as x-->infinity.

I(x)=\int^{\infty}_{0}e^{-xt} ln(1+\sqrt{t})dt

Homework Equations

The Attempt at a Solution

I'm not exactly sure how to begin with this. I've read through the section on my book on asymptotics but am still confused. I think I need to expand the integrand as a Taylor Series.

 

[STEMucator]

Do you remember the Taylor series for e^x and ln(1+x)?

 

[the_kid]

Do you remember the Taylor series for e^x and ln(1+x)?

Yes.

e^{x}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots

ln(1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\cdots

 

[STEMucator]

Yes.

e^{x}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots

ln(1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\cdots

Write them as sums for ease, remember though that the series for ln(1+x) does not exist at n=0 so start your series at n=1.

Now how can you re-write your integral so you can integrate term by term?

 

[the_kid]

Write them as sums for ease, remember though that the series for ln(1+x) does not exist at n=0 so start your series at n=1.

Now how can you re-write your integral so you can integrate term by term?

How do I know where to truncate the Taylor series so that I can multiply them together?

 

[STEMucator]

So :

e^{-xt} = - \sum_{n=0}^{∞} \frac{xt}{n!}

What happens when you evaluate the first sum? Can you re-write your sum then :)?

And :

ln(1 + \sqrt{t}) = \sum_{n=1}^{∞} (-1)^{n+1} \frac{t^{n/2}}{n}

 

[the_kid]

So :

e^{-xt} = - \sum_{n=0}^{∞} \frac{xt}{n!}

What happens when you evaluate the first sum? Can you re-write your sum then :)?

And :

ln(1 + \sqrt{t}) = \sum_{n=1}^{∞} (-1)^{n+1} \frac{t^{n/2}}{n}

Hmm, I'm not exactly sure what you mean. Could you clarify?

 

[STEMucator]

Hmm, I'm not exactly sure what you mean. Could you clarify?

What is the first term of the series of e^-xt

 

[the_kid]

What is the first term of the series of e^-xt

I believe it is 1...

 

[STEMucator]

I believe it is 1...

Oh a bit of a fumble earlier. Sorry it's 2am.

e^{-xt} = \sum_{n=0}^{∞} \frac{(-xt)^n}{n!}

If the first term is 1, then re-write :

e^{-xt} = 1 + \sum_{n=1}^{∞} \frac{(-xt)^n}{n!}

 

[the_kid]

Oh a bit of a fumble earlier. Sorry it's 2am.

e^{-xt} = \sum_{n=0}^{∞} \frac{(-xt)^n}{n!}

If the first term is 1, then re-write :

e^{-xt} = 1 + \sum_{n=1}^{∞} \frac{(-xt)^n}{n!}

No worries!

OK, I've got that. I'm just not sure where to go from there.

 

[STEMucator]

Multiply the two sums of the integral together and then integrate term by term.

 

[the_kid]

Multiply the two sums of the integral together and then integrate term by term.

OK, simple enough. It looks like this series diverges when x tends to infinity, though? Is that expected?

 

[the_kid]

OK, simple enough. It looks like this series diverges when x tends to infinity, though? Is that expected?

Thinking about this, it seems to make sense. When x goes to infinity, the e^(-xt) becomes small quickly, and so the integral is dominated by the log term--but integrating that term from 0 to infinity diverges, so it makes sense that the expansion would diverge as x tends to infinity. Right?

 

[STEMucator]

I'm not sure, I didn't actually compute it, but by the looks of it I believe the integral will diverge.

 

[the_kid]

Multiply the two sums of the integral together and then integrate term by term.

How exactly do I multiply these two together if they have infinite terms?