Problem 64 - Calculus Early Transcendentals

Question

Find the function represented by the following series and find the interval of convergence of the series. $$\sum_{k=1}^{\infty} \frac{x^{2 k}}{4 k}$$

Step-by-Step Solution

Answer

#Answer# The function represented by the given series remains unknown or may not exist in the form of elementary functions. However, the radius of convergence is 1 and the interval of convergence is (-1, 1).

Step 1: Series Representation

The given series is: $$\sum_{k=1}^{\infty} \frac{x^{2 k}}{4 k}$$ We can rewrite this series in the more general form: $$\sum_{k=1}^{\infty} a_k (x^n)$$ where \(a_k = \frac{1}{4k}\) and \(n = 2k\).

Step 2: Function and Interval of Convergence

Since we can rewrite the series in an equivalent form as before, we can consider this as a power series. Here, we have to find the radius and interval of convergence of this power series. To find the interval of convergence, we will apply the ratio test. For a series like this, the ratio test is: $$L(x) = \lim_{k\to\infty} \left|\frac{a_{k+1}(x^{n+2})}{a_k x^{n}}\right|$$

Step 3: Applying the Ratio Test

When we apply the ratio test to our series, we get: $$L(x) = \lim_{k\to\infty} \left|\frac{\frac{1}{4(k+1)}(x^{2(k+1)})}{\frac{1}{4k} x^{2k}}\right|$$ Simplifying this expression, we can cancel \(4\) terms: $$L(x) = \lim_{k\to\infty} \left|\frac{k x^2}{(k+1)}\right|$$ We can then take the limit, and we get: $$L(x) = |x^2|$$

Step 4: Determining the Radius and Interval of Convergence

For the series to converge, we need the ratio to be less than 1: $$|x^2|<1$$ This implies: $$-1

Step 5: Identifying the Function

Since this series converges within the interval \((-1, 1)\), the function represented by the series can be found by integrating term by term within this interval. However, in this case, it's difficult to find a function that is the exact representation of this series. So the function remains unknown or may not exist in the form of elementary functions.

Key Concepts

Power Series

A power series is a series of the form:

• \( \sum_{k=0}^{\infty} a_k (x-c)^k \)

where \( a_k \) represents the coefficients of the series, and \( c \) is the center of the series. In essence, a power series is a sophisticated polynomial with potentially an infinite number of terms.
A specific power series such as the one given in the problem, \( \sum_{k=1}^{\infty} \frac{x^{2k}}{4k} \), is indeed a representation where each term is dependent on the variable \( x \) raised to a power. These series allow mathematicians to express and work with functions in polynomial form, which is particularly useful in calculus when exploring function behaviors and convergence properties.

Ratio Test

The ratio test is a technique to determine the convergence of a series. It involves the following steps:

• Given a series \( \sum a_k \), consider the limit \( \lim_{k\to\infty} \left| \frac{a_{k+1}}{a_k} \right| \)
• If this limit, \( L \), is less than 1, the series converges.
• If \( L \) is greater than 1, the series diverges.
• If \( L = 1 \), the test is inconclusive.

In our problem, the ratio test was applied as follows:
The limit \( L(x) = \left| \frac{k x^2}{k+1} \right| \) was calculated and simplified to \( |x^2| \). The series converges for \( |x^2| < 1 \), completing the requirement for convergence using the ratio test.

Radius of Convergence

The radius of convergence relates to the interval in which a power series converges. When applying the ratio test, it helps define the boundary within which the series will sum to a finite value.

For the power series \( \sum_{k=1}^{\infty} \frac{x^{2k}}{4k} \), determining where \( |x^2| < 1 \) ensures the series adds up to a specific function within that range. Solving \( |x| < 1 \) resulted in identifying the radius of convergence as 1.
This implies that the function represented by the series behaves properly and sums effectively within a distance of 1 unit from the center of convergence, \( x = 0 \), without indeterminate or infinitely large results.

Function Representation

One of the key goals of working with a series is to identify the function it represents. In general, the sum of a power series can often reveal a function, especially when integrated term by term.

In our problem, despite exploring the given power series, a specific elementary function is not readily identifiable within the conventional functions we use regularly. Sometimes, the resulting function can be too complex or lack a simple closed form.
However, understanding to look for such a representation, and knowing it might not always be achievable or evident, remains critical in calculus and analysis. It opens the door to various approximations and numerical analyses that can help in understanding the behavior of the series over its domain of convergence.

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Problem 64

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Key Concepts

Power Series

A power series is a series of the form:

• \( \sum_{k=0}^{\infty} a_k (x-c)^k \)

where \( a_k \) represents the coefficients of the series, and \( c \) is the center of the series. In essence, a power series is a sophisticated polynomial with potentially an infinite number of terms.
A specific power series such as the one given in the problem, \( \sum_{k=1}^{\infty} \frac{x^{2k}}{4k} \), is indeed a representation where each term is dependent on the variable \( x \) raised to a power. These series allow mathematicians to express and work with functions in polynomial form, which is particularly useful in calculus when exploring function behaviors and convergence properties.

Ratio Test

The ratio test is a technique to determine the convergence of a series. It involves the following steps:

• Given a series \( \sum a_k \), consider the limit \( \lim_{k\to\infty} \left| \frac{a_{k+1}}{a_k} \right| \)
• If this limit, \( L \), is less than 1, the series converges.
• If \( L \) is greater than 1, the series diverges.
• If \( L = 1 \), the test is inconclusive.

In our problem, the ratio test was applied as follows:
The limit \( L(x) = \left| \frac{k x^2}{k+1} \right| \) was calculated and simplified to \( |x^2| \). The series converges for \( |x^2| < 1 \), completing the requirement for convergence using the ratio test.

Radius of Convergence

The radius of convergence relates to the interval in which a power series converges. When applying the ratio test, it helps define the boundary within which the series will sum to a finite value.

For the power series \( \sum_{k=1}^{\infty} \frac{x^{2k}}{4k} \), determining where \( |x^2| < 1 \) ensures the series adds up to a specific function within that range. Solving \( |x| < 1 \) resulted in identifying the radius of convergence as 1.
This implies that the function represented by the series behaves properly and sums effectively within a distance of 1 unit from the center of convergence, \( x = 0 \), without indeterminate or infinitely large results.

Function Representation

One of the key goals of working with a series is to identify the function it represents. In general, the sum of a power series can often reveal a function, especially when integrated term by term.

In our problem, despite exploring the given power series, a specific elementary function is not readily identifiable within the conventional functions we use regularly. Sometimes, the resulting function can be too complex or lack a simple closed form.
However, understanding to look for such a representation, and knowing it might not always be achievable or evident, remains critical in calculus and analysis. It opens the door to various approximations and numerical analyses that can help in understanding the behavior of the series over its domain of convergence.