Answer
The interval of convergence for the series is: $(-\infty, +\infty)$ and the radius of convergence is $R=+ \infty$.
Work Step by Step
Ratio Test: Let us consider a series $\Sigma a_k$ whose limit $l$ can be obtained as: $ l=\lim\limits_{k \to \infty} |\dfrac{a_{k+1}}{a_k}|$
1. For $l \lt 1$, the series is absolutely convergent.
2. For $l \gt 1$, the series is divergent.
3. For $l = 1$, the series is inconclusive.
Therefore, $ L=\lim\limits_{k \to \infty} |\dfrac{a_{k+1}}{a_k}|\\=\lim\limits_{k \to \infty} |\dfrac{\pi^{k+1} (x-1)^{2k+2}}{(2k+3)!} \times \dfrac{(2k+1)!}{\pi^k (x-1)^{2k}}| \\=0$
So, we can conclude that the given series converges absolutely if $l =0 \lt 1$ . This means that the series converges for all real numbers.
Therefore, the interval of convergence for the series is: $(-\infty, +\infty)$ and the radius of convergence is $R=+ \infty$.
