I know that there are many variants of a Taylor Series, but the one I am most intrigued in is this Taylor Series: $$\gamma+\ln|x|+\sum_{n=1}^{\infty}\frac{x^{n}}{(n!)(n)} $$I am mostly confused as to how they derived the gamma constant.
This is what I tried to do, but it didn't work (Edit: See below, as this is complete nonsense): $$\int_{}^{}\frac{e^{x}}{x}dx=\int_{}^{}\sum_{n=0}^{\infty}\frac{x^{n-1}}{n!}dx=\int_{}^{}\frac{1}{x}+\sum_{n=1}^{\infty}\frac{x^{n-1}}{n!}dx=-\frac{1}{x^{2}}+\sum_{n=1}^{\infty}\frac{x^{n}}{(n!)(n)}$$ Edit: My above expression is completely wrong as I derivated 1/x rather than actually taking the real antideriative of 1/x which is ln|x|. Also I realized that that original taylor series could have been modified to include the absolute value. So we have to correct derivation here: $$\int_{}^{}\frac{1}{x}+\sum_{n=1}^{\infty}\frac{x^{n-1}}{n!}dx=C+\ln|x|+\sum_{n=1}^{\infty}\frac{x^{n}}{(n!)(n)}$$ Now in this case, I am still confused as to how to derive the C constant as being the Euler-Macheroni Constant.
