Answer
$T(x)=1+2x+2x^2+\dfrac{4x^3}{3}+......+\dfrac{(2x)^n}{n!}+.......$
Work Step by Step
Here, we have: $f(c)=1 \\ f'(c)=2 \\ f''(c)=4 \\ f'''(c)=8$
This leads to $f^n (c)=2^n$
Next, we can write as: $a_n=\dfrac{f^n (c)}{n!}=\dfrac{2^n}{n!}$
Hence, the required Taylor series is:
$T(x)=1+2x+2x^2+\dfrac{4x^3}{3}+......+\dfrac{(2x)^n}{n!}+.......$
