# Help with Recurrence Formula

1. May 12, 2012

### QCM~

Hi guys. I'm having a bit of trouble with what I thought was a simple math question.

1. The problem statement, all variables and given/known data
$x_{n}$ = $\int_0^1 \frac{t^n}{t+7}dt$

Show that $x_0$=ln(8/7) and $x_n = n^{-1} - 7x_{n-1}$

2. The attempt at a solution

Showing x0 = ln(8/7) is a vanilla textbook log question. I'm having trouble with the second part. I am using integration by parts on the form:
$\int^1_0 t \frac{t^{n-1}}{t+7}dt$
and letting u=t and dv=$\frac{t^{n-1}}{t+7}dt$
This give:
$tx_{n-1}|^1_0 - \int_0^1 x_{n-1} dt\\ = x_{n-1} - \int_0^1 x_{n-1} dt$

at which point I'm stuck. I'm not sure if I've used the right IBP substitution or if I'm just almost there and it's just a case of simplifying what I have into a more general case (but can't see it).

Thanks

2. May 12, 2012

### cjc0117

I'm still trying to figure it out using IBP, but try using long division on the integrand instead to arrive at a recursive relationship for the quotient. Then integrate.

Last edited: May 13, 2012
3. May 12, 2012

### Dick

Very astute! I don't think integration by parts leads anywhere. You might change your post to just the hint, "try long division" instead of giving the whole solution. It's more subtle.

4. May 12, 2012

### cjc0117

Thanks Dick. And yes, I suppose you're right. I just edited it.