Find the interval of convergence of the series $ \sum^{\infty}\limits_{k=0} ((-1)^k+3)^k(x-1)^k $ I wish to find the interval of convergence of the following series
\begin{align} \sum^{\infty}_{k=0} ((-1)^k+3)^k(x-1)^k \end{align}
PROOF
Wittingly,
\begin{align} \left[(-1)^k+3\right]^k= \begin{cases} 0,&\text{if}\;j=0;\\4^j,&\text{if}\;j=2k;\\2^j,&\text{if}\;j=2k+1.\end{cases}  \end{align}
Thus,
\begin{align} \limsup_{k\to\infty} \sqrt[k]{\left|((-1)^k+3)^k(x-1)^k\right|}&=|x-1|\limsup_{k\to\infty}\sqrt[k]{ 2^{2k+1}}\\&=|x-1|\limsup_{k\to\infty} 2^{(2k+1)\times \frac{1}{k}} \\&=4|x-1| \end{align}
The series converges absolutely when $\limsup_{k\to\infty} \sqrt[k]{\left|((-1)^k+3)^k(x-1)^k\right|}<1,$ i.e., 
\begin{align} |x-1|< \dfrac{1}{4}\iff \dfrac{3}{4}<x<\dfrac{5}{4} \end{align}
QUESTION:
Why must \begin{align} \limsup_{k\to\infty} \sqrt[k]{\left|((-1)^k+3)^k(x-1)^k\right|}= \limsup_{k\to\infty} \sqrt[k]{\left| 2^{2k+1}(x-1)^k\right|}\end{align}
as stated in the book before me and not
\begin{align} \limsup_{k\to\infty} \sqrt[k]{\left|((-1)^k+3)^k(x-1)^k\right|}= \limsup_{k\to\infty} \sqrt[k]{\left| 4^{2k}(x-1)^k\right|}\;?\end{align}

Your justification of the equality$$\limsup_{k\to\infty}\sqrt[k]{((-1)^k+3)^k(x-1)^k}=\lvert x-1\rvert\limsup_{k\to\infty}\sqrt[k]{ 2^{2k+1}}$$doesn't hold. You have $\bigl((-1)^{2k+1}+3\bigr)^{2k+1}=2^{2k+1}$. Therefore, what you should consider here is$$\sqrt[2k+1]{2^{2k+1}}=2.$$But, since $\bigl((-1)^{2k}+3\bigr)^{2k}=4^{2k}$ and$$\sqrt[2k]{4^{2k}}=4,$$you have$$\limsup_{k\to\infty}\sqrt[k]{((-1)^k+3)^k(x-1)^k}=4\lvert x-1\rvert.$$
