91Ó°ÊÓ

First, let's rewrite the power series \(f(x)\) in terms of \(f(ax)\): $$ f(ax) = \sum_{k=0}^{\infty} c_k (ax)^k = \sum_{k=0}^{\infty} (a^k c_k) x^k $$ Now we have the power series for \(f(ax)\) in the form of \(\sum (a^k c_k) x^k\).

For the power series \(f(x) = \sum c_k x^k\) with \(|x|<R\), we know that $$ \lim_{k \to \infty} \left|\frac{c_{k+1}x^{k+1}}{c_{k}x^{k}}\right| = \frac{|x|}{R} < 1. $$ For the power series \(f(ax) = \sum (a^k c_k)x^k\), we need to apply the Ratio Test again: $$ \lim_{k \to \infty} \left|\frac{(a^{k+1} c_{k+1})x^{k+1}}{(a^k c_{k})x^{k}}\right| = \lim_{k \to \infty} |a|\left|\frac{c_{k+1}x^{k+1}}{c_kx^k}\right| $$ Now, let's substitute the result from the original power series into the equation for \(f(ax)\): $$ \lim_{k \to \infty} |a|\left|\frac{c_{k+1}x^{k+1}}{c_kx^k}\right| = |a| \cdot \frac{|x|}{R} < 1. $$ To find the interval of convergence for \(f(ax)\), we need to solve the inequality \(|a| \cdot \frac{|x|}{R} < 1\) for \(|x|\):

Solve the inequality for \(|x|\): $$ |a| \cdot \frac{|x|}{R} < 1 \Rightarrow |x| < \frac{R}{|a|} $$ Now we have the interval of convergence for the power series \(f(ax)\): \(|x|<\frac{R}{|a|}\).

The interval of convergence for the power series \(f(ax)\), where \(a\) is a non-zero real number, is \(|x|<\frac{R}{|a|}\).