I cannot understand where the red term "k" below comes from?
Why is that "k" in red?
$f(x)=\displaystyle\sum_{k=0}^{\infty}B_k\frac{x^k}{k!}\ \wedge \ e^x=\displaystyle\sum_{k=0}^{\infty}\frac{x^k}{k!}$
$f(x)e^x=(\displaystyle\sum_{k=0}^{\infty}B_k\frac{x^k}{k!})(\displaystyle\sum_{k=0}^{\infty}\frac{x^k}{k!})=$
$=\displaystyle\sum_{k=0}^{\infty}(\displaystyle\sum_{i=0}^{k}B_i\frac{x^i}{i!}\frac{x^{k-i}}{(k-i)!})=\displaystyle\sum_{k=0}^{\infty}(\displaystyle\sum_{i=0}^{k}\frac{B_i}{i!(k-i)!})x^k=\displaystyle\sum_{k=0}^{\infty}(\displaystyle\sum_{i=0}^{k}\binom{k}{i}B_i)\frac{x^k}{k!}$
$c_k=\displaystyle\sum_{i=0}^{k}\binom{k}{i}B_i=\displaystyle\sum_{i=0}^{k-1}\binom{k}{i}B_i+B_k=\color{red}k\color{black}+B_k$
$f(x)e^x=\displaystyle\sum_{k=0}^{\infty}(k+B_k)\frac{x^k}{k!}=\displaystyle\sum_{k=1}^{\infty}\frac{x^k}{(k-1)!}+f(x)=xe^x+f(x)$
$f(x)=\frac{xe^x}{e^x-1}\ \wedge \ e^x>0 \ \Rightarrow f(x)=\frac{x}{1-e^{-x}}$
Should there be "x" instead of "k" because of equation below?
$n>1\ \Rightarrow \displaystyle\sum_{k=0}^{n-1}\binom{n}{k}B_k=0$
$\frac{x}{e^x-1}=\displaystyle\sum_{i=0}^{\infty}\frac{B_ix^i}{i!}$
$n=1 \Rightarrow \displaystyle\sum_{k=0}^{0}$$\binom{1}{0}B_0\frac{x}{1!}=x$