Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

Prove Theorem D as follows: Let $$ f(x)=1+\sum_{n=1}^{\infty}\left(\begin{array}{l} p \\ n \end{array}\right) x^{n} $$ (a) Show that the series converges for \(|x|<1\). (b) Show that \((1+x) f^{\prime}(x)=p f(x)\) and \(f(0)=1\). (c) Solve this differential equation to get \(f(x)=(1+x)^{p}\).

Step-by-Step Solution

Answer

The series converges for \(|x|<1\); the solution is \(f(x)=(1+x)^p\).

Step 1: Analyzing Series Convergence

The series given is \( f(x) = 1 + \sum_{n=1}^{\infty} \binom{p}{n} x^n \). To show convergence for \(|x| < 1\), we can use the ratio test. The coefficients \(\binom{p}{n}\) are such that for large \(n\), \(\binom{p}{n} = \frac{p(p-1)...(p-n+1)}{n!}\). For the ratio test, calculate the limit: \[\lim_{n \to \infty} \left| \frac{\binom{p}{n+1} x^{n+1}}{\binom{p}{n} x^n} \right| = \lim_{n \to \infty} \left| \frac{(p-n)x}{n+1} \right| = |x|.\] This limit is less than 1 for \(|x| < 1\), hence the series converges for \(|x| < 1\).

Step 2: Deriving the Differential Equation

First, differentiate \(f(x)\) to find \(f'(x)\). Each term differentiates as \(f'(x) = \sum_{n=1}^{\infty} n \binom{p}{n} x^{n-1}\). From the binomial coefficient property \(n \binom{p}{n} = p \binom{p-1}{n-1}\), substitute into the derivative: \(f'(x) = \sum_{n=1}^{\infty} p \binom{p-1}{n-1} x^{n-1}\). Re-index to align powers of \(x\): set \(m = n-1\), giving \(f'(x) = p \sum_{m=0}^{\infty} \binom{p-1}{m} x^m = pf(x)/(1+x)\). Simplifying gives \((1+x)f'(x) = pf(x)\).

Step 3: Initial Condition Inspection

Evaluate \(f(0)\) to find the constant in the solution. Because the series starts with 1 when \(x=0\) and there are no other terms with zero power, \(f(0) = 1\). This matches the initial condition provided.

Step 4: Solving the Differential Equation

To solve the differential equation \((1+x)f'(x) = pf(x)\), rewrite as \(\frac{f'(x)}{f(x)} = \frac{p}{1+x}\). Integrate both sides with respect to \(x\): on the left, the integral is \(\ln f(x)\); on the right, the integral becomes \(p\ln(1+x)\). Thus, \(\ln f(x) = p\ln(1+x) + C\), where \(C\) is a constant. Exponentiating both sides yields \(f(x) = e^C(1+x)^p\). Use the initial condition \(f(0)=1\) to solve for \(e^C\): \(1 = e^C \cdot 1^p\), so \(e^C = 1\), giving the solution \(f(x) = (1+x)^p\).

Key Concepts

Series Convergence

The objective is to establish the convergence of the series \( f(x) = 1 + \sum_{n=1}^{\infty} \binom{p}{n} x^n \) for \(|x| < 1\). Series convergence is crucial to determine when an infinite series approaches a finite value. For this, we employed the ratio test—a popular method to find convergence radius of series.

This involves calculating the limit \( \lim_{n \to \infty} \left| \frac{\binom{p}{n+1} x^{n+1}}{\binom{p}{n} x^n} \right| \). Here the binomial coefficients \( \binom{p}{n} \) become important, which roughly behave as \( \frac{p(p-1)...(p-n+1)}{n!} \) for large \(n\). The simplification of this expression leads to the conclusion \(|x| < 1\) ensures that the series converges.

This fundamental step establishes that for values of \(x\) within the interval \(-1, 1\), the series reaches desired limits, making subsequent calculations valid and applicable.

Differential Equations

Differential equations describe relationships between functions and their derivatives. They are fundamental in modeling and solving problems involving rates of change.

In our context, we were given the challenge of showing that \((1+x)f'(x) = pf(x)\). By differentiating \(f(x)\), we obtained a series representation of \(f'(x)\). Subsequent algebraic manipulation, utilizing the property \( n \binom{p}{n} = p \binom{p-1}{n-1} \), enables transformation of the series into the form \(pf(x)/(1+x)\), equating to \(f'(x)\).

This result gives rise to the differential equation \((1+x)f'(x) = pf(x)\), which is pivotal in transitioning from an infinite series to a clear explicit solution. Solving these equations is key in transforming theoretical mathematical challenges into functioning equations describing real-world phenomena.

Initial Conditions

Initial conditions provide the necessary information to find a unique solution to a differential equation. They specify the value of the function at a certain point—essentially, they are starting points for solutions.

For this problem, the series \( f(x) \) was inspected at \( x=0 \) to consider the value of the function, resulting in \( f(0) = 1 \). This value stems from the fact that every term with \(x\) vanishes except the constant term. By confirming \( f(0) = 1 \), we meet a critical requirement needed to determine a constant involved when integrating the differential equation.

Initial conditions hence play a deciding role in defining the particular solution from a set of possible solutions given by the original differential equation. Without establishing these, the result remains indefinite, even if convergence and other properties are understood.

Binomial Theorem

The binomial theorem is a crucial algebraic principle providing a formula for expanding powers of sums. Expressed as \((1+x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k\), it allows expanding expressions involving power sums.

This theorem links directly to our differential equation solution. After solving the differential equation, we arrived at \( f(x) = (1+x)^p \), a clear result of applying the binomial theorem. The theorem's utility manifests through its expansion, explaining the form of our function \( f(x) \) and confirming our series results.

Overall, the binomial theorem simplifies complex algebraic processes by providing predictable terms in series expansions. When confirming expressions stemming from differential equations or similar mathematical problems, this understanding is indispensable.