Question

Write out the sum of the first 5 terms of the given power series. $$\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n) !} x^{2 n}$$

Short answer

1 - \frac{1}{2} x^{2} + \frac{1}{24} x^{4} - \frac{1}{720} x^{6} + \frac{1}{40320} x^{8}

Step-by-step solution

Understand the Power Series Formula

The given power series is \( \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n)!} x^{2 n} \). This formula represents an infinite sum starting from \( n=0 \). Each term in the series can be expanded as \( \frac{(-1)^{n}}{(2n)!} x^{2n} \). To find the sum of the first 5 terms, we will calculate terms from \( n = 0 \) to \( n = 4 \).

Calculate the First Term

For \( n = 0 \), the term is \( \frac{(-1)^{0}}{(2\times0)!} x^{2\times0} = \frac{1}{0!} x^{0} = 1 \), since \( x^{0} = 1 \) and \( 0! = 1 \).

Calculate the Second Term

For \( n = 1 \), the term is \( \frac{(-1)^{1}}{(2\times1)!} x^{2\times1} = \frac{-1}{2!} x^{2} = \frac{-1}{2} x^{2} \), since \( 2! = 2 \).

Calculate the Third Term

For \( n = 2 \), the term is \( \frac{(-1)^{2}}{(2\times2)!} x^{2\times2} = \frac{1}{4!} x^{4} = \frac{1}{24} x^{4} \), since \( 4! = 24 \).

Calculate the Fourth Term

For \( n = 3 \), the term is \( \frac{(-1)^{3}}{(2\times3)!} x^{2\times3} = \frac{-1}{6!} x^{6} = \frac{-1}{720} x^{6} \), since \( 6! = 720 \).

Calculate the Fifth Term

For \( n = 4 \), the term is \( \frac{(-1)^{4}}{(2\times4)!} x^{2\times4} = \frac{1}{8!} x^{8} = \frac{1}{40320} x^{8} \), since \( 8! = 40320 \).

Summarize the First Five Terms

The sum of the first five terms is: \( 1 - \frac{1}{2} x^{2} + \frac{1}{24} x^{4} - \frac{1}{720} x^{6} + \frac{1}{40320} x^{8} \).

Key concepts

Summation

Summation refers to the process of adding a sequence of numbers together. In the context of power series, we're often looking at an infinite summation where terms involve a formula depending on some variable. For example, in our exercise, the power series is given by \( \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n)!} x^{2 n} \). Each term in this sum is expressed and calculated according to this formula.

When computing a sum, especially for infinite series, you often start by computing a finite number of terms. This helps you understand the behavior of the series or estimate its value. Here, we're focused on only the first five terms. The sum of these terms gives us a partial view of the larger infinite series, providing insight into how these summations work.

Factorials

Factorials are a fundamental concept in mathematics often denoted with an exclamation mark \(!\). The factorial of a non-negative integer \(n\), written as \(n!\), is the product of all positive integers less than or equal to \(n\). For example, \(4! = 4 \times 3 \times 2 \times 1 = 24\). When \(n = 0\), we define \(0! = 1\).

Factorials are significant in power series, especially those involving permutations and combinations, as they appear in the denominators of the terms. In our exercise, the terms involve even factorials like \((2n)!\). This means for \(n=0,1,2,3,4\), we need to compute \((2\times0)!, (2\times1)!, (2\times2)!, \) and so on, resulting in values like 1, 2, 24, and more.

Series Expansion

Series expansion refers to expressing a function as a series, often a power series, involving terms of powers of a variable. In mathematics, this is a powerful technique to approximate functions. The given series in the exercise can be seen as an expansion of a function in terms of powers of \(x\).

For example, the terms of the series after expansion are in the form \(\frac{(-1)^n}{(2n)!}x^{2n}\). This allows us to approximate the behavior of the function using a finite number of terms, namely the first five terms we've calculated. In real-world applications, such series expansions are useful because they provide simpler polynomial forms that approximate more complex functions.

Mathematics

Mathematics is the foundation of all sciences. It helps us model real-world phenomena, solve problems, and understand the intricacies of patterns and structures. In the exercise about power series, we're exploring concepts that are fundamental in calculus and analysis.

By breaking down the power series into specific terms, students get an opportunity to see the interplay between summation, factorials, and the structure of a mathematical expansion. Making sense of seemingly complex infinite processes becomes manageable. It's the mathematical ideas and algorithms such as these that solve everyday problems, contribute to technological advancements, and deepen our appreciation of the world around us. It shows how extended sums can be applied in everything from physics to economics, ensuring that mathematics remains an invaluable tool for both education and practical life.