I was working doing some integrals for my project and i get stuck in how they use the partial fraction to obtain the following
$$ \dfrac{1}{(x+1)(x+2)(x+3)\cdots (x+n)}=\sum_{k=1}^{n} \dfrac{c_k}{x+k} $$ where $c_k=\dfrac{\pi^{-k}}{k}$.
Can anyone help me with this? Or give me some ideas in how to obtain it.
Thank you in advance for your help.
The correct expression for $c_k$ is:
$$c_k=\frac{1}{\prod\limits_{1\leq i\leq n,\,i\ne k}(i-k)}$$
This can be obtained most easily from the so-called "cover up method" for partial fractions.
To see all the steps:
$$\frac{1}{(x+1)\cdots(x+n)} = \frac{c_1}{x+1}+\cdots+\frac{c_n}{x+n}$$
Multiply both sides by the denominator on the left:
$$1 = \sum_{k=1}^n \left(c_k\prod_{1\leq i\leq n,\,i\ne k}(x+i)\right)$$
Setting $x=-k$, every term on the right except for the $k$th term becomes $0$, so we have:
$$1=c_k\prod_{1\leq i\leq n,\,i\ne k}(-k+i)$$
This leads directly to our solution.