Question

Powers of \(x\) multiplied by a power series Prove that if \(f(x)=\sum_{k=0}^{\infty} c_{k} x^{k}\) converges with radius of convergence \(R,\) then the power series for \(x^{m} f(x)\) also converges with radius of convergence \(R,\) for positive integers \(m\)

Short answer

Short Answer: The radius of convergence for the power series \(x^m f(x)\) is the same as the radius of convergence for the original function \(f(x)\) because when applying the ratio test to the new power series, the additional factor of \(|x|^m\) does not alter the inequalities that determine the radius of convergence, thus maintaining the same value of \(R\).

Step-by-step solution

Recall the definition of radius of convergence

The radius of convergence (\(R\)) of a power series is the value for which the series converges absolutely for all \(|x| < R\) and diverges for all \(|x| > R\). In order to find the radius of convergence, we can use the ratio test, where if the limit of the ratio between consecutive terms approaches a value smaller than 1, the series converges.

Write down the power series for \(x^m f(x)\)

Now, let's find the power series for \(x^m f(x)\). It can be obtained by multiplying the given power series by \(x^m\) as follows: $$ x^m f(x) = x^m \sum_{k=0}^{\infty} c_k x^k = \sum_{k=0}^{\infty} c_k x^{k+m}. $$

Apply the ratio test to the new power series

In order to find the radius of convergence for the new power series, we can apply the ratio test by finding the limit of the ratio between consecutive terms as the index \(k\) goes to infinity: $$ \lim_{k \to \infty} \frac{|c_{k+1}x^{(k+1)+m}|}{|c_k x^{k+m}|} = \lim_{k \to \infty} \frac{|c_{k+1}|x^{k+1+m}}{|c_k|x^{k+m}}. $$ To simplify this expression, we can cancel out \(x^k\) from both numerator and denominator: $$ \lim_{k \to \infty} \frac{|c_{k+1}|}{|c_k|} |x|^m. $$

Compare the ratio test result to the original series

Now, as the given series converges with radius of convergence \(R\), we already know from the ratio test for the original series: $$ \lim_{k \to \infty} \frac{|c_{k+1}|}{|c_k|} |x| < 1 $$ for \(|x| < R\) and \(|x| > R\) for divergence. Comparing this expression to the expression we found for the new power series, we see that the additional factor of \(|x|^m\) in the new series does not change the inequality when \(|x| < R\), so the new power series also converges for \(|x| < R\). Likewise, if we consider \(|x| > R\), the additional factor of \(|x|^m\) will only increase the value of the expression, meaning that it still diverges for \(|x| > R\).

Conclusion

Since the power series for \(x^m f(x)\) converges for \(|x| < R\) and diverges for \(|x| > R\), we can conclude that the radius of convergence for \(x^m f(x)\) is also \(R\), proving our claim.

Key concepts

Power Series

A power series is a way of expressing a function as an infinite sum of terms that involve powers of a variable, usually denoted as \(x\). In mathematical terms, a power series takes the form \(\sum_{k=0}^{\infty} c_k x^k\), where \(c_k\) are coefficients based on a particular function.

• Each term in a power series is dependent on \(x\), the variable, and \(k\), the index that increases incrementally.
• The power series serves as a tool to represent complex functions as simpler polynomial approximations.

Understanding a function through its power series allows us to analyze characteristics of the function, such as its points of convergence and the behavior around these points. Power series can represent functions over intervals where they become useful in calculations and approximations.
To analyze a power series, we often need to determine its radius of convergence—this is where the series converges, providing a valid representation of the function for the values of \(x\) within that range.

Ratio Test

The ratio test is a method used in calculus to test the convergence of an infinite series, particularly a power series. This test evaluates the limit of the ratio of consecutive terms. It's described by the following expression:\[\lim_{k \to \infty} \frac{|c_{k+1}x^{k+1}|}{|c_k x^k|}\]

• If this limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive on its own. Other tests might be needed.

The ratio test is especially useful when analyzing power series because it helps determine their radius of convergence. As we multiply each term by a higher power of \(x\), the test examines how the series behaves, highlighting whether it will converge or diverge.
When applying the ratio test to a modified power series like \(x^m f(x)\), you determine if the new terms' sequence still maintains the convergence condition of the original series.

Convergence

Convergence in the context of a power series refers to whether the sum of its terms approaches a finite value as more terms are added. Specifically, a power series converges if there exists an interval or region where the series’ expression of a function comes to a stable value.

• The notion of convergence ensures a function can be expressed accurately within a series form for given values of \(x\).
• The point at which the series starts diverging, no longer leading to a stable sum, defines the boundary of convergence, often described by the radius of convergence \(R\).

When we multiply a power series by \(x^m\), the convergence behavior remains unchanged because the multiplication affects all terms equally. It's important to note that the core condition for convergence still depends on whether \(|x| < R\), consistent with the original series.
The convergence analysis also involves checking a series under boundary conditions, \(|x| > R\), to confirm that the series diverges beyond the radius of convergence. This coherent behavior guarantees that the manipulation of a series, like multiplying by \(x^m\), does not alter the range over which the original series was convergent.