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Chapter 6: Problem 111

Suppose that the coefficients \(a_{n}\) of the series \(\sum_{n=0}^{\infty} a_{n} x^{n}\) are defined by the recurrence relation \(a_{n}=\frac{a_{n-1}}{n}+\frac{a_{n-2}}{n(n-1)} . \quad\) For \(\quad a_{0}=0 \quad\) and \(\quad a_{1}=1,\) compute and plot the sums \(S_{N}=\sum_{n=0}^{N} a_{n} x^{n}\) for \(N=2,3,4,5\) on [-1,1].

Short Answer

Expert verified
Calculate coefficients up to \(a_5\), then compute \(S_N\) for each \(N\), and plot them on [-1,1].

Step by step solution

01

Calculate coefficients using recurrence relation

Start with the initial conditions given: \(a_0 = 0\) and \(a_1 = 1\). Apply the recurrence relation \(a_n = \frac{a_{n-1}}{n} + \frac{a_{n-2}}{n(n-1)}\) to find subsequent coefficients. For \(n=2\), \(a_2 = \frac{a_1}{2} + \frac{a_0}{2(1)} = \frac{1}{2}\). For \(n=3\), \(a_3 = \frac{a_2}{3} + \frac{a_1}{3(2)} = \frac{1}{6} + \frac{1}{6} = \frac{1}{3}\). Continuing this process: \(a_4 = \frac{a_3}{4} + \frac{a_2}{4(3)} = \frac{1}{12} + \frac{1}{24} = \frac{1}{8}\), and \(a_5 = \frac{a_4}{5} + \frac{a_3}{5(4)} = \frac{1}{40} + \frac{1}{60} = \frac{1}{24}\).
02

Compute sums for given N values

Using the coefficients calculated, compute \(S_N = \sum_{n=0}^{N} a_n x^n\) for \(N=2, 3, 4, 5\). For \(N=2\), \(S_2 = a_0 + a_1x + a_2x^2 = 0 + x + \frac{1}{2}x^2\). For \(N=3\), \(S_3 = S_2 + a_3x^3 = x + \frac{1}{2}x^2 + \frac{1}{3}x^3\). For \(N=4\), \(S_4 = S_3 + a_4x^4 = x + \frac{1}{2}x^2 + \frac{1}{3}x^3 + \frac{1}{8}x^4\). For \(N=5\), \(S_5 = S_4 + a_5x^5 = x + \frac{1}{2}x^2 + \frac{1}{3}x^3 + \frac{1}{8}x^4 + \frac{1}{24}x^5\).
03

Plot each polynomial \(S_N\) on the interval [-1,1]

Use a graphing tool or software to plot each polynomial \(S_2, S_3, S_4, S_5\) over the interval \([-1,1]\). Ensure each polynomial is graphed to visualize how the sum of the series approaches as \(N\) increases. Label each graph according to its respective polynomial and notice the changes in curve shape as more terms are added.

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Key Concepts

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

Recurrence Relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values. Each term is defined as a function of its predecessors. In the exercise, the recurrence relation \( a_n = \frac{a_{n-1}}{n} + \frac{a_{n-2}}{n(n-1)} \) helps us calculate the coefficients \( a_n \) of the power series.
  • A recurrence relation provides an efficient way to compute terms based on previous values without doing extensive calculations for each element.
  • The initial conditions \( a_0 = 0 \) and \( a_1 = 1 \) are vital as they serve as the starting point for the recursive computation. Without them, we couldn't generate the sequence of coefficients.
  • Using this recurrence relation, each subsequent term involves simple operations on the preceding terms, making it computationally feasible and straightforward to find \( a_2, a_3, a_4, \) etc.
Power Series
A power series is an infinite series of the form \( \sum_{n=0}^{\infty} a_n x^n \), where \( a_n \) are coefficients, and \( x \) is a variable. The main idea is to express complex functions as sums of simpler polynomial terms.
  • Computing power series using recurrence relations can simplify finding approximations for more involved functions.
  • The power series can cover a range of values for \( x \) and converge to a specific function within a radius of convergence.
  • In this exercise, by calculating partial sums \( S_N = \sum_{n=0}^{N} a_n x^n \), you approximate the value of the power series up to a chosen degree, offering insights into the behavior of the entire series.
Polynomial Approximation
Polynomial approximation involves using polynomial expressions to represent more complex functions. This method is handy because polynomials are easier to manipulate and work with analytically.
  • The Polynomials \( S_2, S_3, S_4, \) and \( S_5 \) approximate the curve of the power series by only considering a finite number of terms.
  • These approximations can give a clear idea of how the function behaves within a specific interval, in this case, \([-1, 1]\).
  • When more terms are added, the accuracy of polynomial approximation typically increases, especially within the interval where the series converges.
Taylor Series
The Taylor series is a special type of power series that approximates a function using derivatives at a single point. It generalizes the idea of representing functions as infinite sums.
  • The Taylor series can be seen as a way to create polynomial approximations of smooth functions near a specific point \( x = a \).
  • In practical applications, truncating the Taylor series at a certain degree leads to an approximation polynomial, resembling the construction of \( S_N \) polynomials in the exercise.
  • Although this problem doesn't explicitly construct a Taylor series, the concept aligns with the creation of partial sums of power series to approximate functions gradually.
Graphing Polynomials
Graphing polynomials is an essential technique for visualizing function behavior in a specified range, such as \([-1, 1]\). Visualization helps understand how polynomial approximations change with more terms.
  • By plotting \( S_2, S_3, S_4, \) and \( S_5 \), you can see how each polynomial behaves and how closely they match as approximations to the original function over the interval.
  • Graphical representation helps determine if adding more terms leads to diminishing returns or continues to refine the approximation within the desired range.
  • It's a crucial step for verifying the accuracy and convergence of the polynomial approximation to the actual series or function.

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Most popular questions from this chapter

Compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of \(f\). $$ f(x)=\frac{\tan \sqrt{x}}{\sqrt{x}} \text { (see expansion for } \left.\tan x\right) $$

Suppose that \(y=\sum_{k=0}^{\infty} a_{k} x^{k}\) satisfies \(y^{\prime}=-2 x y\) and \(y(0)=0 .\) Show that \(a_{2 k+1}=0\) for all \(k\) and that \(a_{2 k+2}=\frac{-a_{2 k}}{k+1} .\) Plot the partial sum \(S_{20}\) of \(y\) on the interval [-4,4].

Suppose that a set of standardized test scores is normally distributed with mean \(\mu=100\) and standard deviation \(\sigma=10\). Set up an integral that represents the probability that a test score will be between 90 and 110 and use the integral of the degree 10 Maclaurin polynomial of \(\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}\) to estimate this probability.

Compute the Taylor series of each function around \(x=1\). $$ f(x)=\frac{1}{x} $$

Find the Maclaurin series of each function. $$ f(x)=2^{x} $$
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