Question

Determine whether the following statements are true and give an explanation or counterexample. a. The interval of convergence of the power series \(\Sigma c_{k}(x-3)^{k}\) could be (-2,8) b. \(\sum(-2 x)^{k}\) converges, for \(-\frac{1}{2} < x < \frac{1}{2}\) c. If \(f(x)=\sum c_{k} x^{k}\) on the interval \(|x|<1\), then \(f\left(x^{2}\right)=\sum c_{k} x^{2 k}\) on the interval \(|x|<1\) d. If \(f(x)=\sum c_{k} x^{k}=0,\) for all \(x\) on an interval \((-a, a),\) then \(c_{k}=0,\) for all \(k\)

Short answer

Based on the given step-by-step solution, the following short answer can be written: For the power series \(\Sigma c_{k}(x-3)^{k}\), its interval of convergence could be (-2, 8), since it is possible but cannot be guaranteed without knowing the coefficients \(c_k\). The series \(\sum(-2 x)^{k}\) converges for \(-\frac{1}{2} < x < \frac{1}{2}\). If \(f(x) = \sum c_{k} x^{k}\) on the interval \(|x| < 1\), then \(f(x^2) = \sum c_{k} x^{2k}\) on the interval \(|x| < 1\). If a power series converges to zero on an open interval and has at least two non-zero coefficients, then all coefficients must be zero due to the Uniqueness Theorem.

Step-by-step solution

Analyze the statement\

The given power series is \(\sum_{k=0}^{\infty} c_{k}(x-3)^{k}\). In order to find out if its interval of convergence could be (-2, 8), we apply the Ratio Test. \

Apply Ratio Test\

To apply the ratio test, we examine the limit $$\lim_{k\to\infty}\left|\frac{c_{k+1}(x-3)^{k+1}}{c_{k}(x-3)^{k}}\right| = (x-3) \lim_{k\to\infty} \left|\frac{c_{k+1}}{c_{k}}\right|.$$ The ratio test states that if the limit is less than 1, then the series converges on that interval; if the limit is greater than 1, then it diverges. \

Determine convergence for the given interval\

Since the given interval is (-2, 8), we have \(|x-3| < 5\), and we want to know if the series converges when \(|(x-3)| < 5\). We cannot determine the limit without knowing the coefficients \(c_k\), so we cannot say for sure if the power series converges on this interval. Therefore, the statement is true because it is possible for the interval of convergence to be (-2, 8). #b. \(\sum(-2 x)^{k}\) converges, for \(-\frac{1}{2} < x < \frac{1}{2}\)# \

Analyze the statement\

The given power series is \(\sum_{k=0}^{\infty} (-2x)^{k}\). Now, we have to find out if this series converges on the interval \(-\frac{1}{2} < x < \frac{1}{2}\). \

Apply the Ratio Test\

To apply the ratio test, we examine the limit $$\lim_{k\to\infty}\left|\frac{-2x(x)^{k+1}}{-2x(x)^k}\right| = |x|.$$ According to the Ratio Test, the series converges when the limit is less than 1. So, we have $$|x| < 1.$$ \

Check the given interval for convergence\

The condition \(|x| < 1\) is indeed satisfied in the interval \(-\frac{1}{2} < x < \frac{1}{2}\). Therefore, the statement is true. #c. If \(f(x)=\sum c_{k} x^{k}\) on the interval \(|x|<1\), then \(f\left(x^{2}\right)=\sum c_{k} x^{2 k}\) on the interval \(|x|<1\)# \

Analyze the statement\

The question asks if we can substitute \(x^2\) for \(x\) in the power series and still maintain the convergence interval. \

Substitute \(x^2\) in the power series\

Given that \(f(x) = \sum_{k=0}^{\infty} c_{k}x^{k}\), we substitute \(x^2\) into the power series to find $$f(x^2) = \sum_{k=0}^\infty c_{k}(x^{2})^{k} = \sum_{k=0}^\infty c_{k}x^{2k}.$$ \

Check the convergence interval\

Since the original power series converges for \(|x| < 1\), when we substitute \(x^2\) for \(x\), we still have \(|x^2| < 1\), which implies \(|x| < 1\). Therefore, the new power series converges on the same interval, and the statement is true. #d. If \(f(x)=\sum c_{k} x^{k}=0,\) for all \(x\) on an interval \((-a, a),\) then \(c_{k}=0,\) for all \(k\)# \

Analyze the statement\

Given that the power series is equal to zero on the interval \((-a, a)\), we need to find out if all coefficients \(c_k\) are equal to zero. \

Apply the Uniqueness Theorem\

The Uniqueness Theorem states that if there is a power series that converges to zero on an open interval and has at least two non-zero coefficients, then all the coefficients must be zero. This means that if \(f(x) = \sum_{k=0}^{\infty} c_{k} x^{k} = 0\) for all \(x\) on the interval \((-a, a)\), then all coefficients \(c_k\) must be equal to zero. Therefore, the statement is true.

Key concepts

Interval of Convergence

When working with power series, the **interval of convergence** is incredibly important because it tells us over which range of values the series will converge to a real number. This interval is found by applying various convergence tests, with the Ratio Test being the most commonly used tool.
In simple terms, the interval of convergence is the range of x-values for which the series converges. So if a power series is \[ \sum c_{k}(x-a)^{k} \], the series converges when values of \(x\) are within a certain distance from \(a\).
Determining Interval with Example:

• Consider the series \( \Sigma c_{k}(x-3)^{k} \)
• Using the Ratio Test, if it shows convergence within \(-2 < x < 8\), then this is the interval of convergence.
• End values need separate checks to see if they are included in the interval. This is because convergence may not hold exactly at the endpoints.

Understanding intervals of convergence helps in knowing where power series can act like regular functions.

Ratio Test

The **Ratio Test** is a powerful method used to determine the convergence of an infinite series. It's particularly useful for power series due to its easy application to series terms involving polynomials and powers.
Here’s a simple breakdown of how you apply the Ratio Test:

• For a series \( \sum_{k=0}^{\infty} a_{k} \), consider the limit
  \[ \lim_{k\to\infty} \left| \frac{a_{k+1}}{a_{k}} \right| \]
• If this limit is less than 1, the series converges.
• If the limit is greater than 1, the series diverges.
• If it equals 1, the test is inconclusive.

This method is frequently used owing to its simplicity — you only compare consecutive terms. This allows us to easily find the interval of convergence by testing the series with this test in our defined interval.

Uniqueness Theorem

The **Uniqueness Theorem** for power series is an important concept, particularly in understanding series that equate to zero. This theorem states that if a power series converges to zero on an entire interval, under conditions allowing convergence beyond just zero itself, then every coefficient of the series must be zero.
Conceptual Insights:

• A power series \( f(x) = \sum_{k=0}^{\infty} c_{k} x^{k} \) equates to zero.
• If converging to zero over an interval \, \((-a, a)\), then all coefficients \(c_k\) must be zero.
• This means a power series can't be zero on an interval unless it is the zero series itself.

Understanding this theorem helps when analyzing whether unique series representations can exist, especially when dealing with possible "flat" series (ones equating identically to zero).

Convergence Interval

The concept of a **convergence interval** refers to the specific span or interval within which a power series remains convergent. It highlights the boundaries of x-values where the series representation of a function properly behaves and gives real or complex numbers, as intended.
Key Points to Consider:

• The length and center of the convergence interval form the base of the series' applicability.
• If you have \( |x - a| < R \):
  • \(a\) is the center.
  • \(R\) (radius) defines how far out you can go from \(a\).
• It's crucial for understanding where the series serves as an accurate representation of the function it describes.
• Beyond this interval, you must evaluate the endpoints separately, because series could converge incorrectly when exactly reaching the boundaries.

Having a defined convergence interval helps in many practical calculations and analytic considerations for what theoretical solutions a series might imply.