Please answer the following questions for $T(n).$

$T(n)=2+4+8+16+$cdots$+2^n$

$(1)$ With the help of the following summation formula of geometric series, simplify $T(n).$

Geometric series: $)(1$

Special case: $x<mn\; id="EditorElement\_911491">1</mn>$:$,{\displaystyle \underset{k=0}{\overset{}{}}}{x}^k=\frac{1}{1-x}$

$(2)$ Use the induction method to prove the result you get for question $(1)$ is correct for any $n1.$

$(3)$ Based on the result you get for question $(1),$ obtain correct $$ notation for $T(n).$

$(4)$ Optional: To obtain the asymptotic notation of $T(n),$ can you drop all the lower order terms in $T(n)?$