/*! 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 19 Find a scries expansion for \(\i... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 19

Find a scries expansion for \(\int_{0}^{x} e^{-t^{2}} d t\) for \(x \in \mathbb{R}\).

Short Answer

Expert verified
The series expansion of \(\int_{0}^{x} e^{-t^{2}} dt\) is \(\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!(2n+1)} x^{2n+1}\) for \(x \in \mathbb{R}\).

Step by step solution

01

Power series of \(e^{x}\)

The exponential function \(e^{x}\) can be expanded as a power series: \(e^{x} = \sum_{n=0}^{\infty} \frac{x^{n}}{n!}\). Now, simply substitute \(x\) with \(-t^{2}\) to get the power series of \(e^{-t^{2}}\): \(e^{-t^{2}} = \sum_{n=0}^{\infty} \frac{(-t^{2})^{n}}{n!}.\)
02

Integrate the power series

Now, integrate the power series term by term from 0 to \(x\): \(\int_{0}^{x} e^{-t^{2}} dt = \int_{0}^{x} \sum_{n=0}^{\infty} \frac{(-t^{2})^{n}}{n!} dt = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!} \int_{0}^{x} t^{2n} dt\).
03

Find the integral of \(t^{2n}\)

The integral of \(t^{2n}\) from 0 to \(x\) is \(\int_{0}^{x} t^{2n} dt = \frac{x^{2n+1}}{2n+1}\).
04

Substitute Step 3 into Step 2

Substitute the integral of \(t^{2n}\) from Step 3 into the formula from Step 2: \(\int_{0}^{x} e^{-t^{2}} dt = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!(2n+1)} x^{2n+1}\).

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Key Concepts

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

Power Series
A power series is a way of expressing a function as an infinite sum of terms. Each term is in the form of a constant multiplied by a variable raised to a successive power. Power series can represent complex functions in simpler polynomials, which makes them extremely useful for analysis and calculations.
  • A power series centered at a point \( a \) generally looks like this: \( \sum_{n=0}^{\infty} c_n (x-a)^n \), where \( c_n \) are coefficients.
  • Power series converge to the function within a certain range (radius of convergence), meaning the series accurately represents the function within this interval.
  • The exponential function \( e^x \) is a classic example of a power series: \[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]
  • By substituting values into the variable, you can manipulate the power series to fit specific needs, like in this problem, where \( x \) was replaced with \( -t^2 \) to obtain the power series for \( e^{-t^2} \).
Understanding power series is crucial because these allow complex functions to be expressed and manipulated more easily using algebraic operations.
Exponential Function
The exponential function, denoted by \( e^x \), is a mathematical constant raised to the power of a variable. It plays a significant role in mathematics, particularly in calculus, differential equations, and complex analysis.
  • The base \( e \) is an irrational number approximately equal to 2.71828.
  • The exponential function has a few notable properties: it is always greater than zero, its rate of growth increases with the value of \( x \), and it has a constant rate of proportional increase.
  • One of the main features of the exponential function is its power series expansion: \( e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \). This series expansion converges for all real numbers \( x \) due to the factorial \( n! \) in the denominator, which grows very quickly.
  • In our exercise, we explore a special case where the exponential function is composed with a quadratic function (\(-t^2\)). The power series approach allows us to evaluate integrals involving \( e^{-t^2} \), which do not have elementary antiderivatives.
The exponential function's flexibility in combination with its power series representation is powerful for solving a wide array of mathematical problems.
Integration of Series
Integrating a series involves evaluating the integral term by term. This is particularly useful when dealing with functions that cannot be easily integrated using standard calculus techniques.
  • When a function is expressed as a power series, you can integrate it by integrating each term of the series individually.
  • For a power series \( \sum_{n=0}^{\infty} a_n x^n \), its integral from 0 to \( x \) is \( \sum_{n=0}^{\infty} \frac{a_n}{n+1} x^{n+1} \).
  • This technique is applicable as long as the series converges uniformly on the interval of integration.
  • In our exercise, the series for \( e^{-t^2} \) is integrated from 0 to \( x \), providing \( \int_{0}^{x} \sum_{n=0}^{\infty} \frac{(-t^{2})^{n}}{n!} dt = \sum_{n=0}^{\infty} \frac{(-1)^n}{n!(2n+1)} x^{2n+1} \).
  • Term-by-term integration simplifies calculations and helps construct series solutions for integrals that are otherwise intractable using elementary functions.
Integration of series opens up avenues for solving more complex integration problems that appear in fields like physics, engineering, and beyond.

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Most popular questions from this chapter

Consider the series $$1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}-\frac{1}{7}++--\cdots$$ where the signs come in pairs. Does it converge?

Determine the radius of convergence of the series \(\sum a_{n} x^{n}\), where \(a_{n}\) is given by: (a) \(1 / n^{n}\). (b) \(n^{\alpha} / n !\), (c) \(n^{n} / n !\), (d) \((\ln n)^{-1}, \quad n \geq 2\), (e) \((n !)^{2} /(2 n) !\) (f) \(n^{-\sqrt{n}} .\)

If \(a\) and \(b\) are positive numbers, then \(\sum(a n+b)^{-p}\) converges if \(p>1\) and diverges if \(p \leq 1\).

Suppose that none of the numbers \(a, b, c\) is a negative integer or zero. Prove that the hypergeometric series $$\frac{a b}{1 ! c}+\frac{a(a+1) b(b+1)}{2 ! c(c+1)}+\frac{a(a+1)(a+2) b(b+1)(b+2)}{3 ! c(c+1)(c+2)}+\cdots$$ is absolutely convergent for \(c>a+b\) and divergent for \(c

Suppose that \(\sum a_{n}\) is a convergent series of real numbers. Either prove that \(\sum b_{n}\) converges of give a counter-example, when we define \(b_{n}\) by (a) \(a_{n} / n\). (b) \(\sqrt{a_{n}} / n \quad\left(a_{n} \geq 0\right)\). (c) \(a_{n} \sin n\), (d) \(\sqrt{a_{n} / n} \quad\left(a_{n} \geq 0\right)\), (e) \(n^{1 / n} a_{n}\), (f) \(a_{n} /\left(1+\left|a_{n}\right|\right)\).
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