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Chapter 6: Problem 2

By integration, differentiation, or any other valid operation, find the functions which are given by the following power series. (a) \(\sum_{i}^{\infty} n^{2} x^{n}\) (b) \(\sum_{0}^{\infty} \frac{x^{n}}{(2 n) !}\) (c) \(\sum_{0}^{\infty} \frac{x^{2 n+1}}{2 n+1}\) (d) \(x+x^{4}+x^{7}+x^{10}+x^{13}+\cdots\)

Short Answer

Expert verified
(a) \(\frac{2x}{(1-x)^3}\), (b) \(\frac{1}{2}e^{x^2}\), (c) \(2\ln|1+x^2|\), (d) \(\frac{x}{1-x^3}\)

Step by step solution

01

Function for Series (a)

Rewrite the summation \(\sum_{i}^{\infty} n^{2} x^{n}\) as \(x \sum_{i}^{\infty} n^{2} x^{n-1}\). This series is the derivative of \(\sum_{i}^{\infty} n x^{n-1}\) which is \(x \sum_{i}^{\infty} n x^{n-1}\), the derivative of \(\sum_{i}^{\infty} x^n\). This sum converges for \(-1 < x < 1\) and equals \(\frac{1}{1-x}\). Two times differentiation yields \(\frac{2}{(1-x)^3}\) multiply by \(x\) to get \( \frac{2x}{(1-x)^3}\).
02

Function for Series (b)

Recognize that the series \(\sum_{0}^{\infty} \frac{x^{n}}{(2 n) !}\) is similar to the Taylor series for the exponential function (if only Evens examined). By doubling every term, this sum can be related to \(e^{x}\). Thus, \(\sum_{0}^{\infty} \frac{x^{n}}{(2 n) !} = \frac{1}{2}e^{x^2}\).
03

Function for Series (c)

The series \(\sum_{0}^{\infty} \frac{x^{2 n+1}}{2 n+1}\) can be related to \(\sum_{0}^{\infty} \frac{x^{n}}{ n}\), which is the natural logarithm series for \(1+x\), taking only odd powered terms. This gives us \(2\ln|1+x^2|\).
04

Function for Series (d)

The sequence of powers in \(x+x^{4}+x^{7}+x^{10}+x^{13}+\cdots\) forms a geometric sequence with common ratio \(x^{3}\). The sum of a geometric series \(\frac{a}{1-r}\) where \(a= x\), and \(r= x^3\). Therefore, the sum of the series is \(\frac{x}{1-x^3}\).

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Key Concepts

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

Integration
Integration is a fundamental concept in calculus. It involves finding a function whose derivative matches the integrand (the function being integrated). Power series can sometimes be solved by integration, especially when trying to find an antiderivative.
For example, in series (b) from the exercise, we're looking at
  • The terms are similar to parts of the Taylor series for the exponential function, except only even powers are present.
  • By recognizing this pattern, we can understand the series as being a component of the exponential function's series expansion where integration helps us uncover relationships to known functions like e^{x} .
Understanding integration in the context of power series can help make sense of how series relate to elementary functions and can be instrumental in finding closed-form solutions.
Differentiation
Differentiation is the mathematical process of finding the derivative of a function. It's like finding the rate at which something changes. Power series, which are infinite sums of terms in the form of polynomials, can sometimes be differentiated to find simpler or more recognizable forms.
For instance, in series (a),
  • We derived a new function by differentiating twice: starting with a simple geometric series, differentiating yields the series for individual polynomial terms raised to powers.
  • This process reflects the use of known differentiation rules applied to power series, resulting in a more familiar function that describes the series' sum within a certain interval of convergence.
Differentiating power series is valuable because it can transform complex series into functions that are easier to interpret or relate to other known functions.
Taylor Series
Taylor series allow us to express functions as an infinite sum of terms calculated from the values of a function鈥檚 derivatives at a single point. This concept is crucial because it gives us a powerful tool to approximate and understand complex functions using polynomials.
For series (b),
  • The terms resemble those in the Taylor expansion of the exponential function \( e^x \).
  • By focusing only on even-powered terms of the typical expansion, the exercise illustrates how a relationship to a known series (like the exponential function) can simplify understanding the entire expression.
  • Recognizing Taylor series patterns enables you to link infinite series to existing mathematical properties and functions you already understand well.
Taylor series are the backbone for approximating functions and can simplify working with power series by connecting them to functions we're already familiar with.
Convergence
Convergence pertains to whether an infinite series approaches a finite number as more terms are added. Understanding convergence is essential to ensure that power series behave predictably and deliver usable results.
Exploring series within their convergence intervals is necessary:
  • For series (a), we determined that it converges for \(-1 < x < 1\), which guides us on where the function derived from the series behaves correctly.
  • Such information ensures calculations are based on limits where the function remains valid and prevents errors from arising in extrapolating results.
Being aware of convergence conditions helps you make sure that the results obtained from power series are mathematically sound and applicable within the specified ranges, thereby enhancing reliability in practical applications.

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Most popular questions from this chapter

Show that \(\int_{1}^{\infty} d x \int_{1}^{\infty} G(x, y) d y \neq \int_{1}^{\infty} d y \int_{1}^{\infty} G(x, y) d x\) if \(G(x, y)=(x-y) /(x+y)^{3}\).

Let \(F(x)=1+2 x+x^{2}+2 x^{3}+x^{4}+2 x^{5}+\cdots \cdot\) By division, find a power series expansion near \(x=0\) for \(1 / F(x)\)

If \(f(x)=a_{1} x+a_{2} x^{2}+\cdots\), then \(1 / f(x)=1-a_{1} x+\left\\{\left(a_{1}\right)^{2}-a_{2}\right\\} x^{2}+\left\\{2 a_{1} a_{2}-\left(a_{1}\right)^{3}-a_{3}\right\\} x^{3}\) \(+A x^{4}+\cdots\) Find \(A\)

Show that there is no function \(g(t)\) such that \(\int_{0}^{\infty} g(t) d t\) converges but such that \(x^{2} t e^{-x t} \leq g(t)\) for all \(x \geq 1, t \geq 0\)

Investigate the existence and uniformity of $$ \lim _{t \rightarrow 0} \frac{x \sin (x t)}{1+x^{2}} $$
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