Problem 60
Question
If the power series \(f(x)=\sum c_{k} x^{k}\) has an interval of convergence of
\(|x|
Step-by-Step Solution
Verified Answer
Answer: The interval of convergence for the power series $f(ax)$ is $|x|<\frac{R}{|a|}$.
1Step 1: Define the power series for f(ax)
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 \(\sum (a^k c_k) x^k\).
2Step 2: Apply the Ratio Test to find the interval of convergence for f(ax)
For the power series \(f(x) = \sum c_k x^k\) with \(|x|
3Step 3: Find the interval of convergence for f(ax)
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 \(f(ax)\): \(|x|<\frac{R}{|a|}\).
4Step 4: Write the final answer
The interval of convergence for the power series \(f(ax)\), where \(a\) is a non-zero real number, is \(|x|<\frac{R}{|a|}\).
Other exercises in this chapter
Problem 60
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