##\left(1+\dfrac1n\right)^n=e^{n\ln\left(1+\dfrac1n\right)}## and using Taylor expansion ##\ln\left(1+\dfrac1n\right)=\dfrac 1n-\dfrac1{2n^2}+\dfrac1{3n^3}-\cdots##
##n\ln\left(1+\dfrac1n\right)=1-\dfrac 1{2n}+\dfrac1{3n^2}-\cdots##
##e^{n\ln(1+\frac1n)}=e.e^{-\frac1{2n}+\frac 1{3n^2}-...}##...