<s>[INST] 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}
 [/INST] Hint
$$
\sum^{n}_{k=0} ((-1)^k+3)^k(x-1)^k =\lim_{n\to \infty}\left(\sum_{k=0}^n 4^{2k}(x-1)^{2k}+2(x-1)\sum_{k=0}^n 2^{2k}(x-1)^{2k}\right)
$$
and now
$$
\sum_{k=0}^n 4^{2k}(x-1)^{2k} = \frac{1-4^{2(n+1)}(x-1)^{2(n+1)} }{1-4^{2}(x-1)^{2}}
$$
which converges if $|4(x-1)| < 1$ etc. so the result is
$$
\sum^{n}_{k=0} ((-1)^k+3)^k(x-1)^k=\frac{2(x-1)}{1-2^2 (x-1)^2}+\frac{1}{1-4^2 (x-1)^2} 
$$
for $\frac 34\lt x\lt \frac 54$
Attached the plot showing in red the sum approximation for $n = 10$ and in black the asymptotic result found

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