/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 Find the sum of the series. $$... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 63

Find the sum of the series. $$\sum_{n=0}^{\infty}(-1)^{n} \frac{X^{4 n}}{n !}$$

Short Answer

Expert verified
The sum of the series is \( e^{-X^4} \).

Step by step solution

01

Identify the Series Type

Observe that the given series is in the form \( \sum_{n=0}^{\infty} (-1)^n \frac{X^{4n}}{n!} \). This series resembles the Maclaurin series for the exponential function \( e^x \), specifically the series \( e^{-x} = \sum_{n=0}^{\infty} \frac{(-x)^n}{n!} \).
02

Substitute and Simplify

Substitute \( x = X^4 \) into the known series for \( e^{-x} \), which gives: \( e^{-X^4} = \sum_{n=0}^{\infty} \frac{(-X^4)^n}{n!} = \sum_{n=0}^{\infty} (-1)^n \frac{X^{4n}}{n!} \).
03

Match the Series

Notice that these two series are exactly the same, which confirms our substitution is correct. It matches the given series \( \sum_{n=0}^{\infty} (-1)^n \frac{X^{4n}}{n!} \).
04

Provide the Sum of the Series

Since the given series is equivalent to the series expansion of \( e^{-X^4} \), the sum of the series is \( e^{-X^4} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Series Convergence
When we discuss series convergence, we're focusing on whether the sum of a series approaches a specific value as more terms are added. For a series to be convergent, the partial sums must approach a finite limit. In simpler terms, adding up the terms shouldn't result in a never-ending number; instead, it should settle to a particular number.
  • A series like the one in our step by step solution, \( \sum_{n=0}^{\infty} (-1)^n \frac{X^{4n}}{n!} \), converges if its terms diminish swiftly enough.
  • The convergence is often confirmed by comparing a series to a known convergent series.
Pivotal techniques like the Ratio Test or Absolute Convergence Test are often deployed to judge the convergence of more complex series.
Exponential Function
The exponential function, particularly in the context of calculus, is one of the most significant functions. It's often expressed as \( e^x \), where \( e \) is approximately 2.71828 and is known as Euler's number.
  • The exponential function is central in describing continual growth or decay, such as in populations, radioactive decay, and even financial systems.
  • In calculus, the series for \( e^x \) is represented as \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \), which converges for all \( x \).
Thus, the exponential function's series representation offers a blueprint for constructing and understanding various infinite series, like the one we examined in our solution.
Substitution in Series
Substitution in series is a handy technique where we replace a variable in a known series expansion to derive a related series or solve a problem. Our exercise involved substituting \( x = X^4 \) into the exponential series \( e^{-x} = \sum_{n=0}^{\infty} \frac{(-x)^n}{n!} \).
  • This substitution allowed us to match the given series with the known structure of the exponential function, transforming it into \( e^{-X^4} \).
  • By recognizing this pattern, we simplify complex expressions and directly determine the sum of the series.
This method leverages the power of known series expansions to tackle a broad array of mathematical problems with efficiency.
Infinite Series
An infinite series is an unending sum of elements from a sequence. In mathematical notation, this is represented by the sigma notation \( \sum_{n=0}^{\infty} a_n \), where \( a_n \) represents the sequence's terms.
  • Calculating an infinite series means finding the sum between all terms starting from the initial term to infinity, a process effective when the series converges.
  • A well-known example handled in coursework is the geometric series, which imparts foundational understanding of what it means for a series to converge.
Our exercise was an instance of an infinite series, expressed as \( \sum_{n=0}^{\infty} (-1)^n \frac{X^{4n}}{n!} \), highlighting how infinite series are managed using known patterns or formulas.

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Most popular questions from this chapter

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