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Chapter 5: Problem 23

Rewrite the given expression as a sum whose generic term involves \(x^{n} .\) $$ x \sum_{n=1}^{\infty} n a_{n} x^{n-1}+\sum_{k=0}^{\infty} a_{k} x^{k} $$

Short Answer

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Question: Rewrite the given expression as a sum whose generic term involves \(x^n\): $$ x \sum_{n=1}^{\infty} n a_n x^{n-1} + \sum_{k=0}^{\infty} a_{k} x^{k} $$ Answer: The given expression can be rewritten as: $$ \sum_{j=0}^{\infty} (x (j+1) a_{j+1} + a_{j}) x^{j} $$

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01

Adjust the exponents and indices for both sums

Consider the first sum and let \(m = n - 1\), so \(n = m + 1.\) Now, the first sum can be rewritten as: $$ x \sum_{m=0}^{\infty} (m+1) a_{m+1} x^{m} $$ The second sum remains the same: $$ \sum_{k=0}^{\infty} a_{k} x^{k} $$
02

Combine the two sums using the same index

Now that we have adjusted the exponents and indices in both sums, we can combine the two sums using the same index, \(j\). For the first sum, set \(j=m\). For the second sum, set \(j=k\). Then, the combined sum is: $$ \sum_{j=0}^{\infty}(x (j+1) a_{j+1} x^{j} + a_{j} x^{j}) $$
03

Simplify the sum

Inside the sum, both terms have the common factor \(x^j\). We can rewrite the sum as: $$ \sum_{j=0}^{\infty} (x (j+1) a_{j+1} + a_{j}) x^{j} $$ This expression is a sum with a generic term involving \(x^j\).

Key Concepts

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

Power Series
Power series are an essential concept in calculus and analysis, representing complex functions in a form that's easy to manipulate. Generally, a power series is an infinite series of the form: ewlineewline\[ \text{Power Series:} \quad \text{P}(x) = \text{a}_0 + \text{a}_1x + \text{a}_2x^2 + \text{a}_3x^3 + \text{dots} = \sum_{n=0}^{\text{infty}} \text{a}_nx^n \]ewlinewhere \( \text{a}_n \) represents the coefficient of the nth term, and \( x \) is a variable. This series converges within a certain radius of convergence, meaning it sums up to a specific function value within that interval around the origin.When manipulating power series, such as combining or differentiating them, it’s crucial to keep in mind that operations are valid within their interval of convergence. In the exercise provided, combining the power series by adjusting indices and exponents facilitates simpler expression for further operations like differentiation or integration.
Infinite Series
Infinite series are the cornerstone of understanding mathematical phenomena that unfold over limitless operations.An infinite series refers to the summation of an endless sequence of terms \[ \sum_{n=1}^{\text{infty}} \text{a}_n \]where \( \text{a}_n \) is the nth term of the series. The behavior of infinite series can be particularly complex, as they may converge to a finite limit, diverge to infinity, or oscillate without settling on a specific value.To assess convergence, tests like the ratio test, root test, or comparison test are applied.Moreover, in the context of our exercise, combining two infinite series required careful reindexing to match terms, which is a common manipulation technique in calculus for simplifying expressions or solving complex problems.
Mathematical Induction
Mathematical induction is a powerful proof technique, commonly used to establish the truth of a statement for all natural numbers.The process of mathematical induction involves two critical steps. Firstly, the base case, where we prove that the statement holds for the initial number, often \( n=1 \). Secondly, the inductive step, which shows that if the statement holds for \( n=k \), it also holds for \( n=k+1 \).Although our immediate exercise doesn't directly involve a proof by induction, the principles behind mathematical induction echo in the manipulation of series and sequence expressions. It's an implicit reminder that math often builds upon assumptions that require careful validation through methods like induction.
Calculus
Calculus, the mathematical study of continuous change, is integral in handling various functions and models in mathematics, science, and engineering.It is divided into two main branches: differential calculus, concerning rates of change and slopes of curves; and integral calculus, focusing on the accumulation of quantities and the areas under and between curves.The power and infinite series are foundational in calculus, particularly in Taylor and Fourier series, which express functions as infinite sums of terms calculated from the function's derivatives at a single point.In the given exercise, the concept of combining power series is a prime example of a problem one might encounter in calculus. The manipulation of series, including reindexing and combining terms, is crucial in understanding the behavior of functions and solving more complex calculus problems.

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Most popular questions from this chapter

The Chebyshev equation is $$ \left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+\alpha^{2} y=0 $$ where \(\alpha\) is a constant; see Problem 10 of Section 5.3 . (a) Show that \(x=1\) and \(x=-1\) are regular singular points, and find the exponents at each of these singularities. (b) Find two linearly independent solutions about \(x=1\)

The Bessel equation of order one is $$ x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-1\right) y=0 $$ (a) Show that \(x=0\) is a regular singular point; that the roots of the indicial equation are \(r_{1}=1\) and \(r_{2}=-1 ;\) and that one solution for \(x>0\) is $$ J_{1}(x)=\frac{x}{2} \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(n+1) ! n ! 2^{2 n}} $$ Show that the series converges for all \(x .\) The function \(J_{1}\) is known as the Bessel function of the first kind of order one. (b) Show that it is impossible to determine a second solution of the form $$ x^{-1} \sum_{n=0}^{\infty} b_{n} x^{n}, \quad x>0 $$

Find all singular points of the given equation and determine whether each one is regular or irregular. \(x y^{\prime \prime}+e^{x} y^{\prime}+(3 \cos x) y=0\)

Find all the regular singular points of the given differential equation. Determine the indicial equation and the exponents at the singularity for each regular singular point. \(\left(4-x^{2}\right) y^{\prime \prime}+2 x y^{\prime}+3 y=0\)

The Euler equation \(x^{2} y^{\prime \prime}+\) \(\alpha x y^{\prime}+\beta y=0\) can be reduced to an equation with constant coefficients by a change of the independent variable. Let \(x=e^{z},\) or \(z=\ln x,\) and consider only the interval \(x>0 .\) (a) Show that $$ \frac{d y}{d x}=\frac{1}{x} \frac{d y}{d z} \quad \text { and } \quad \frac{d^{2} y}{d x^{2}}=\frac{1}{x^{2}} \frac{d^{2} y}{d z^{2}}-\frac{1}{x^{2}} \frac{d y}{d z} $$ (b) Show that the Euler equation becomes $$ \frac{d^{2} y}{d z^{2}}+(\alpha-1) \frac{d y}{d z}+\beta y=0 $$ Letting \(r_{1}\) and \(r_{2}\) denote the roots of \(r^{2}+(\alpha-1) r+\beta=0\), show that (c) If \(r_{1}\) and \(r_{2}\) are real and different, then $$ y=c_{1} e^{r_{1} z}+c_{2} e^{r_{2} z}=c_{1} x^{r_{1}}+c_{2} x^{r_{2}} $$ (d) If \(r_{1}\) and \(r_{2}\) are real and equal, then $$ y=\left(c_{1}+c_{2} z\right) e^{r_{1} z}=\left(c_{1}+c_{2} \ln x\right) x^{r_{1}} $$ (e) If \(r_{1}\) and \(r_{2}\) are complex conjugates, \(r_{1}=\lambda+i \mu,\) then $$ y=e^{\lambda z}\left[c_{1} \cos (\mu z)+c_{2} \sin (\mu z)\right]=x^{\lambda}\left[c_{1} \cos (\mu \ln x)+c_{2} \sin (\mu \ln x)\right] $$
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