Problem 73
Question
Find the sum of the series. \( \sum_{n = 0}^{\infty} (-1)^n \frac {x^{4n}}{n!} \)
Step-by-Step Solution
Verified Answer
The sum of the series is \( e^{-x^4} \).
1Step 1: Recognize the Series as a Power Series
The series provided is \( \sum_{n = 0}^{\infty} (-1)^n \frac{x^{4n}}{n!} \). Notice that this resembles the general form of a power series, \( \sum_{n = 0}^{\infty} \frac{a_n(x-x_0)^n}{n!} \). The series has terms containing \((-1)^n\), indicating a potential connection to the exponential series.
2Step 2: Identify the Related Exponential Series
Compare the given series, \( \sum_{n = 0}^{\infty} (-1)^n \frac{x^{4n}}{n!} \), to the exponential series \( e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \). The given series includes \((-1)^n\), suggesting it might be similar to \( e^{-x} \). However, each term in our series is \(x^{4n}\), which means we should consider \( e^{-x^4} \).
3Step 3: Write the Equivalent Exponential Function
The expression \( e^{-x^4} \) can be expanded into a series as \( e^{-x^4} = \sum_{n=0}^{\infty} \frac{(-1)^n (x^4)^n}{n!} = \sum_{n=0}^{\infty} \frac{(-1)^n x^{4n}}{n!} \). Recognize that this is identical to the given series.
4Step 4: Conclude the Sum of the Series
Through the steps, we determined that the infinite series \( \sum_{n = 0}^{\infty} (-1)^n \frac{x^{4n}}{n!} \) is equivalent to \( e^{-x^4} \). Therefore, the sum of the series is \( e^{-x^4} \).
Key Concepts
Exponential SeriesInfinite SeriesSum of Series
Exponential Series
The exponential series is a cornerstone concept in mathematics. It represents the expansion of the exponential function into an infinite series. The classic form of the exponential series is written as \( e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \). Each term in the series contributes to the overall value of the exponential function. This series converges for all real numbers, making it both powerful and versatile in mathematical analysis.
In our exercise, we noticed the series \( \sum_{n=0}^{\infty} (-1)^n \frac{x^{4n}}{n!} \) closely resembles the exponential series form. However, it includes a negative sign \((-1)^n\) and involves powers of \(x^4\). These modifications hint towards a different exponent representation, namely \( e^{-x^4} \), demonstrating how flexible the exponential series structure can be.
In our exercise, we noticed the series \( \sum_{n=0}^{\infty} (-1)^n \frac{x^{4n}}{n!} \) closely resembles the exponential series form. However, it includes a negative sign \((-1)^n\) and involves powers of \(x^4\). These modifications hint towards a different exponent representation, namely \( e^{-x^4} \), demonstrating how flexible the exponential series structure can be.
- Exponential series allows representation of functions as infinite sums.
- It guarantees convergence for all real numbers.
Infinite Series
An infinite series is a sum of an infinite sequence of terms. Each term is obtained through an index, \(n\), which typically starts from zero or one and progresses infinitely. The defining characteristic of an infinite series is its inability to conclusively "finish," because there are always more terms.
Working with infinite series requires understanding convergence. This means determining whether the sum approaches a specific value as more terms are added. In this context, our exercise targets the expression \( \sum_{n = 0}^{\infty} (-1)^n \frac{x^{4n}}{n!} \), an example of an infinite series that uses the factorial \(n!\) to control term growth and ensure convergence.
Working with infinite series requires understanding convergence. This means determining whether the sum approaches a specific value as more terms are added. In this context, our exercise targets the expression \( \sum_{n = 0}^{\infty} (-1)^n \frac{x^{4n}}{n!} \), an example of an infinite series that uses the factorial \(n!\) to control term growth and ensure convergence.
- Infinite series can either converge to a value or diverge.
- They require great care in their manipulation and interpretation.
- Multiplicative factors like \((-1)^n\) introduce alternating series behavior.
Sum of Series
Finding the sum of a series, especially an infinite one, involves determining a closed-form expression that represents the infinite addition of its terms. For power and exponential series, like in our exercise, this involves recognizing patterns that relate to known functions.
Our problem describes a series with a distinct structure \( \sum_{n = 0}^{\infty} (-1)^n \frac{x^{4n}}{n!} \). By comparing it to the exponential series, it can be identified as being equal to \( e^{-x^4} \). This reveals its sum in a compact, function-based form, simplifying an otherwise complex calculation.
Our problem describes a series with a distinct structure \( \sum_{n = 0}^{\infty} (-1)^n \frac{x^{4n}}{n!} \). By comparing it to the exponential series, it can be identified as being equal to \( e^{-x^4} \). This reveals its sum in a compact, function-based form, simplifying an otherwise complex calculation.
- The sum of a series can reveal deeper mathematical relations.
- Patterns in series help in discovering and proving mathematical identities.
- Closed-form expressions represent infinite series efficiently.
Other exercises in this chapter
Problem 72
After injection of a does \( D \) of insulin, the concentration of insulin in patient's system decays exponentially and so it can be written as \( De^{-at}, \)
View solution Problem 72
Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? \( a_n = \cos n \)
View solution Problem 73
When money is spent on goods and services, those who receive the money also spend some of it. The people receiving some of the twice-spent money will spend some
View solution Problem 73
Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? \( a_n = \frac{1}{2n + 3} \)
View solution