Since $f(x)=\ln x$ and $g(x)=e^{x}$ are inverse functions, we know that $e^{\ln (1+x)}=1+x$ for x -1.Find the Taylor series for $e^{\ln (1+x)}$ using only up to the quadratic terms and show that the result is 1 + x.

Question

Since $f(x)=\ln x$ and $g(x)=e^{x}$ are inverse functions, we know that $e^{\ln (1+x)}=1+x$ for x > -1.Find the Taylor series for $e^{\ln (1+x)}$ using only up to the quadratic terms and show that the result is 1 + x.

Quizplus · Accepted Answer

The Answer of Since $f(x)=\ln x$ and $g(x)=e^{x}$ are inverse functions, we know that $e^{\ln (1+x)}=1+x$ for x > -1.Find the Taylor series for $e^{\ln (1+x)}$ using only up to the quadratic terms and show that the result is 1 + x.

Since f(x)=ln⁡xf(x)=\ln xf(x)=lnx And g(x)=exg(x)=e^{x}g(x)=ex Are Inverse Functions, We Know That eln⁡(1+x)=1+xe^{\ln (1+x)}=1+xeln(1+x)=1+x

Since $f(x)=\ln x$ And $g(x)=e^{x}$ Are Inverse Functions, We Know That $e^{\ln (1+x)}=1+x$