Sequence converging to definite integral Let's define
\begin{equation*}
 I_0 := \log\frac{6}{5}
\end{equation*}
and for $k = 1, 2, \ldots, n$
\begin{equation*}
 I_k := \frac{1}{k} - 5 I_{k-1}.
\end{equation*}
How the value $I_n$ is linked with the value of $$\int_0^1\frac{x^n}{x+5} \mathrm{d}x \ ?$$

$I_n+5I_{n-1}=\int_{0}^{1}\frac{x^n}{x+5}dx+5\int_{0}^{1}\frac{x^{n-1}}{x+5}dx=\int_{0}^{1}(\frac{x^n}{x+5}+5\frac{x^{n-1}}{x+5})dx$
$I_n+5I_{n-1}=\int_{0}^{1}x^{n-1}dx=\frac{x^n}{n}|_{0}^{1}=\frac{1}{n}$
