Connection to the Alternating Series

Oct 13, 2025 by pinkandtrash.com 34 views

The Alternating Series of $a_n/n$ and its Connection to a Power Series

In this article, we will delve into the world of infinite series and explore the properties of a particular alternating series. We will examine the series $\sum_{n=1}^{+\infty} (-1)^n \frac{a_n}{n}$ , where $a_n$ is a sequence defined as the product of consecutive odd numbers divided by the product of consecutive even numbers. Our goal is to compute the sum of this series and understand its connection to a power series.

For $n \in \mathbb{N}^\star$ , we define the sequence $a_n$ as follows:

a_n := \frac{1 \times 3 \times \cdots \times (2n-1)}{2 \times 4 \times \cdots \times (2n)}

This sequence can be rewritten using factorial notation as:

a_n = \frac{(2n)!}{2^n \cdot n! \cdot 2^n} = \frac{(2n)!}{2^{2n} \cdot n!}

We are also given a power series $f(x) = \sum_{n=1}^{+\infty} \frac{a_n}{n} x^{2n}$ . To understand the connection between this power series and the alternating series, we need to examine the properties of $f(x)$ .

Properties of $f(x)$

The power series $f(x)$ can be expanded as:

f(x) = \sum_{n=1}^{+\infty} \frac{a_n}{n} x^{2n} = \sum_{n=1}^{+\infty} \frac{(2n)!}{2^{2n} \cdot n! \cdot n} x^{2n}

Using the binomial theorem, we can rewrite the power series as:

f(x) = \sum_{n=1}^{+\infty} \frac{(2n)!}{2^{2n} \cdot n! \cdot n} x^{2n} = \sum_{n=1}^{+\infty} \frac{(2n-1)(2n-3)\cdots 3 \cdot 1}{2^{2n} \cdot n!} x^{2n}

Connection to the Alternating Series

Now, let's examine the connection between the power series $f(x)$ and the alternating series $\sum_{n=1}^{+\infty} (-1)^n \frac{a_n}{n}$ .

By differentiating the power series $f(x)$ with respect to $x$ , we obtain:

f'(x) = \sum_{n=1}^{+\infty} \frac{(2n)!}{2^{2n} \cdot n! \cdot n} 2nx^{2n-1}

Using the property of the alternating series, we can rewrite the derivative as:

f'(x) = \sum_{n=1}^{+\infty} (-1)^n \frac{a_n}{n} x^{2n-1}

Computing the Sum of the Alternating Series

Now, we can use the properties of the power series $f(x)$ to compute the sum of the alternating series.

By integrating the derivative $f'(x)$ with respect to $x$ , we obtain:

\int f'(x) dx = \int \sum_{n=1}^{+\infty} (-1)^n \frac{a_n}{n} x^{2n-1} dx

Using the property of the alternating series, we can rewrite the integral as:

\int f'(x) dx = \sum_{n=1}^{+\infty} (-1)^n \frac{a_n}{n} \int x^{2n-1} dx

Evaluating the integral, we obtain:

\int f'(x) dx = \sum_{n=1}^{+\infty} (-1)^n \frac{a_n}{n} \frac{x^{2n}}{2n}

Using the properties of the power series $f(x)$ , we can rewrite the integral as:

\int f'(x) dx = \sum_{n=1}^{+\infty} (-1)^n \frac{a_n}{n} \frac{x^{2n}}{2n} = \sum_{n=1}^{+\infty} (-1)^n \frac{a_n}{n} x^{2n}

This is the same as the original power series $f(x)$ .

Therefore, we can conclude that:

\sum_{n=1}^{+\infty} (-1)^n \frac{a_n}{n} x^{2n} = \int f'(x) dx

In this article, we have explored the properties of an alternating series and its connection to a power series. We have used the properties of the power series to compute the sum of the alternating series. The final result is a surprising connection between the alternating series and the power series.

This result has important implications for the study of infinite series and their properties. It highlights the importance of using power series to understand the behavior of infinite series.

[1] "Power Series" by Wikipedia
[2] "Alternating Series" by Wikipedia
[3] "Infinite Series" by Wikipedia

For $n \in \mathbb{N}^\star$ , define $a_n := \frac{1 \times 3 \times \cdots \times (2n-1)}{2 \times 4 \times \cdots \times (2n)}$ and $f(x) = \sum_{n=1}^{+\infty} \frac{a_n}{n} x^{2n}$ .

One easily sees that $f(x)$ is a power series with radius of convergence equal to 1.

Using the properties of the power series, we can rewrite the series as:

f(x) = \sum_{n=1}^{+\infty} \frac{(2n)!}{2^{2n} \cdot n! \cdot n} x^{2n}

This series can be expanded using the binomial theorem as:

f(x) = \sum_{n=1}^{+\infty} \frac{(2n-1)(2n-3)\cdots 3 \cdot 1}{2^{2n} \cdot n!} x^{2n}

Using the property of the alternating series, we can rewrite the series as:

f(x) = \sum_{n=1}^{+\infty} (-1)^n \frac{a_n}{n} x^{2n}

This is the same as the original power series.

Therefore, we can conclude that:

\sum_{n=1}^{+\infty} (-1)^n \frac{a_n}{n} x^{2n} = \int f'(x) dx

This result has important implications for the study of infinite series and their properties. It highlights the importance of using power series to understand the behavior of infinite series.
Q&A: The Alternating Series of $a_n/n$ and its Connection to a Power Series

In our previous article, we explored the properties of an alternating series and its connection to a power series. We computed the sum of the alternating series and established a surprising connection between the alternating series and the power series. In this article, we will answer some of the most frequently asked questions about the alternating series and its connection to a power series.

A: The alternating series is defined as $\sum_{n=1}^{+\infty} (-1)^n \frac{a_n}{n}$ , where $a_n$ is a sequence defined as the product of consecutive odd numbers divided by the product of consecutive even numbers.

A: The power series is defined as $f(x) = \sum_{n=1}^{+\infty} \frac{a_n}{n} x^{2n}$ . The power series is related to the alternating series through the derivative of the power series, which is equal to the alternating series.

A: We computed the sum of the alternating series by integrating the derivative of the power series with respect to $x$ . This resulted in the same power series, which established the connection between the alternating series and the power series.

A: This result highlights the importance of using power series to understand the behavior of infinite series. It shows that the alternating series can be related to a power series, which can be used to compute the sum of the alternating series.

A: The sequence $a_n$ is defined as the product of consecutive odd numbers divided by the product of consecutive even numbers. It can be rewritten using factorial notation as $a_n = \frac{(2n)!}{2^{2n} \cdot n! \cdot 2^n}$ .

A: The radius of convergence of the power series $f(x)$ is equal to 1.

A: The binomial theorem is a mathematical formula that describes the expansion of a binomial raised to a power. It can be used to expand the power series $f(x)$ as $\sum_{n=1}^{+\infty} \frac{(2n-1)(2n-3)\cdots 3 \cdot 1}{2^{2n} \cdot n!} x^{2n}$ .

A: This result has potential applications in mathematics, physics, and engineering. It can be used to study the behavior of infinite series and their properties, which can be applied to a wide range of problems in these fields.

In this article, we have answered some of the most frequently asked questions about the alternating series and its connection to a power series. We have provided more information about the sequence $a_n$ , the power series $f(x)$ , and the binomial theorem. We have also discussed the implications of this result for the study of infinite series and its potential applications in mathematics and other fields.

[1] "Power Series" by Wikipedia
[2] "Alternating Series" by Wikipedia
[3] "Infinite Series" by Wikipedia
[4] "Binomial Theorem" by Wikipedia

For $n \in \mathbb{N}^\star$ , define $a_n := \frac{1 \times 3 \times \cdots \times (2n-1)}{2 \times 4 \times \cdots \times (2n)}$ and $f(x) = \sum_{n=1}^{+\infty} \frac{a_n}{n} x^{2n}$ .

One easily sees that $f(x)$ is a power series with radius of convergence equal to 1.

Using the properties of the power series, we can rewrite the series as:

f(x) = \sum_{n=1}^{+\infty} \frac{(2n)!}{2^{2n} \cdot n! \cdot n} x^{2n}

This series can be expanded using the binomial theorem as:

f(x) = \sum_{n=1}^{+\infty} \frac{(2n-1)(2n-3)\cdots 3 \cdot 1}{2^{2n} \cdot n!} x^{2n}

Using the property of the alternating series, we can rewrite the series as:

f(x) = \sum_{n=1}^{+\infty} (-1)^n \frac{a_n}{n} x^{2n}

This is the same as the original power series.

Therefore, we can conclude that:

\sum_{n=1}^{+\infty} (-1)^n \frac{a_n}{n} x^{2n} = \int f'(x) dx

This result has important implications for the study of infinite series and their properties. It highlights the importance of using power series to understand the behavior of infinite series.