Question

Is \(\sum_{k=0}^{\infty} x^{2 k}\) a power series? If so, find the center \(a\) of the power series and state a formula for the coefficients \(c_{k}\) of the power series.

Short answer

If so, what are its center and coefficients? Answer: Yes, the given expression is a power series with center \(a=0\) and coefficients \(c_k = 1\) for \(k = 0,1,2,\dots\) and \(c_k = 0\) otherwise.

Step-by-step solution

Define a power series

A power series is an infinite series of the form: $$\sum_{k=0}^{\infty} c_k(x-a)^k$$ where \(c_k\) are the coefficients, \(a\) is the center and \(x\) is the variable.

Analyze the given series

In our case, the given series is: $$\sum_{k=0}^{\infty} x^{2 k}$$ We want to identify if this is a power series and, if so, determine the center \(a\) and the coefficients \(c_k\).

Identify the center

Observe that the given series can be rewritten as follows: $$\sum_{k=0}^{\infty} (x-a)^{2k},$$ with \(a=0\). This matches the general form of a power series. Therefore, the center of the power series is \(a=0\).

Identify the coefficients

Next, we need to find a formula for the coefficients \(c_k\). Comparing our series with the general form of the power series, we observe the coefficients are associated with the \((x-a)^{2k}\) terms. Specifically, we have: $$c_k = \begin{cases} 1, & k = 0,1,2,\dots \\ 0, & \text{otherwise} \end{cases}$$ Now, we have a formula for the coefficients \(c_k\) of the power series. In conclusion, the given expression \(\sum_{k=0}^{\infty} x^{2 k}\) is a power series with center \(a=0\) and coefficients \(c_k = 1\) for \(k = 0,1,2,\dots\) and \(c_k = 0\) otherwise.

Key concepts

Infinite Series

An infinite series is a sum of an endless sequence of terms. It extends infinitely, which means that it does not terminate after a finite number of steps. When dealing with such series, we often apply mathematical tools and concepts to understand their behavior and characteristics. In our exploration of power series, infinite series is the building block. It consists of terms that keep building upon one another without ever stopping.

Long and complex expressions can seem overwhelming, but breaking them down helps us see the pattern. In our specific exercise, the series \(\sum_{k=0}^{\infty}x^{2k}\) is indeed an infinite series because it suggests adding all the terms generated by \(x^{2k}\), starting from \(k=0\) up into infinity.

Some important aspects of infinite series to remember include:

• The series can sometimes converge to a specific value or expression, providing a precise understanding of its behavior.
• They can diverge, meaning they do not approach a fixed value.
• Recognizing patterns in terms of composition and structure helps in determining the nature of the series.

Coefficients

In power series, coefficients play a crucial role. They are the constants that multiply each term in the series. Understanding the coefficients in a series gives insights into the specific flavor of the series in question. In the general form of a power series, \( \sum_{k=0}^{\infty} c_k (x-a)^k \), the \(c_k\) represents the coefficients of the series.

In our example, the series \(\sum_{k=0}^{\infty}x^{2k}\) has coefficients of 1 for every term where \(k\) is a non-negative integer starting from zero. This is because each term in this series does not change any of the input variable components, but rather relies solely on the exponentiation of \(x\). Therefore, the series can be rewritten in its expanded form as \(1 + x^2 + x^4 + x^6 + \cdots\), with all coefficients being 1. Thus, the coefficients can be seen as:

• \(c_k = 1\) for each term where \(k = 0, 1, 2, ...\)

This consistent presence of 1 as a coefficient indicates that each power of \(x^{2k}\) stands strong on its own without any scaling by another factor.

Center of Power Series

When analyzing a power series, identifying its center is critical. The center is the point around which the series revolves. In mathematical terms, this center is denoted by \(a\) in the standard power series expression \(\sum_{k=0}^{\infty} c_k (x-a)^k\).

The distance and behavior of the series from this center shed light on where the series is most effective or where it converges when it behaves nicely.

In our exercise, the series given is \(\sum_{k=0}^{\infty}x^{2k}\). Notice that the expression \((x-a)^{2k}\) has been used in the solution, and it tells us that \(a=0\) is the center. This means the series is centered at the origin on the x-axis, and it effectively extends its impact symmetrically on both the positive and negative sides of this center.

• Understanding the center helps when graphing or analyzing the behavior near the center.
• At the center, the series representation may be more compact and often usable for approximations or simplifications.

This idea of centering not only makes the mathematical handling of the series convenient but also provides powerful insights in various practical and theoretical applications.