I have the following sequence: $$ y_n = \int_0^1 \frac{x^n}{x+5}\,dx, n = 0,1,\dots $$
Now I have the following recursive algorithm which is equal to the sequence:
$$ y_0 = \log{6} - \log{5} $$ $$ y_n = \frac{1}{n} - 5y_{n-1}, n = 1,... $$
I have to find another recursive algorithm equal to the sequence. I got the tip to write the recursion backwards e.g start with $y_9 \approx 0$
I don't get it, can you help me, thanks.
You already have the equation $y_n = 1/n - 5 y_{n-1}$. But instead of using this to calculate $y_n$ from $y_{n-1}$, you can use it to calculate $y_{n-1}$ from $y_n$. As a starting point take $y_N \approx 0$ where $N$ is fairly large. The fact that this "backwards" recursion is very stable (error is divided by $5$ at each step) means that even if the approximation for $y_N$ is not very good, you will get very good approximations after a few steps.