Is it possible to create a power series for $2^x$ that's always convergent, and where the coefficients are all fractional (that is, bypassing the conversion of $2^x$ to $e^{x\log{2}}$)?
That is, $2^x=\sum_{j=0}^{\infty}c_j x^j$, $\forall$ $x$, where the $c_j$ are fractional.
Please take kindly to this question, I'm a newbie.
No, this is not possible. This would only be possible if $\ln{(2)}$ was rational - which it is not. The method of finding this series expansion can be done avoiding the aforementioned identity by evaluating the function at each of its $n$th derivatives. $$f(0)=2^0=1$$ $$f'(0)=\ln{(2)}\times2^0=\ln{(2)}$$ $$...$$ $$f^{(n)}(0)=\ln^n{(2)}$$ $$f(x)=\sum_{k=0}^\infty \frac{f^{(k)}(0)x^k}{k!}=\sum_{k=0}^\infty \frac{\ln^k{(2)}x^k}{k!}=1+\frac{\ln{(2)}x}{1!}+\frac{\ln^2{(2)}x^2}{2!}+...$$