Question

we indicate how to prove that the sequence \(\left\\{\phi_{n}(t)\right\\},\) defined by Eqs. (4) through (7), converges. Note that $$ \phi_{n}(t)=\phi_{1}(t)+\left[\phi_{2}(t)-\phi_{1}(t)\right]+\cdots+\left[\phi_{x}(t)-\phi_{n-1}(t)\right] $$ (a) Show that $$ \left|\phi_{n}(t)\right| \leq\left|\phi_{1}(t)\right|+\left|\phi_{2}(t)-\phi_{1}(t)\right|+\cdots+\left|\phi_{n}(t)-\phi_{n-1}(t)\right| $$ (b) Use the results of Problem 17 to show that $$ \left|\phi_{n}(t)\right| \leq \frac{M}{K}\left[K h+\frac{(K h)^{2}}{2 !}+\cdots+\frac{(K h)^{n}}{n !}\right] $$ (c) Show that the sum in part (b) converges as \(n \rightarrow \infty\) and, hence, the sum in part (a) also converges as \(n \rightarrow \infty\). Conclude therefore that the sequence \(\left\\{\phi_{n}(t)\right\\}\) converges since it is the sequence of partial sums of a convergent infinite series.

Short answer

Question: Prove the convergence of the sequence \(\left\\{\phi_{n}(t)\right\\}\) using the given expression. Answer: The sequence \(\left\\{\phi_{n}(t)\right\\}\) is convergent since it is the sequence of partial sums of a convergent infinite series derived from applying the triangle inequality and the results from Problem 17. The sum converges to \(e^{K h}\) for all \(K h \in \mathbb{R}\).

Step-by-step solution

Show the inequality with the sum of absolute values

The given expression for \(\phi_n(t)\) is: $$ \phi_{n}(t) \:=\: \phi_{1}(t) \:+\: \left[\phi_{2}(t)-\phi_{1}(t)\right] \:+\: \cdots \:+\: \left[\phi_{n}(t)-\phi_{n-1}(t)\right] $$ Applying the triangle inequality, we have: $$ |\phi_{n}(t)| \:\leq\: |\phi_{1}(t)| \:+\: |[\phi_{2}(t)-\phi_{1}(t)]| \:+\: \cdots \:+\: |[\phi_{n}(t)-\phi_{n-1}(t)]| $$

Use the results of Problem 17

After referring to Problem 17, we get: $$ |\phi_{n}(t) - \phi_{n-1}(t)| \:\leq\: \frac{M}{(n-1)!} (K h)^{n-1} $$ By applying this inequality, we get the following: $$ |\phi_{n}(t)| \:\leq\: \frac{M}{K}\left[K h + \frac{(K h)^{2}}{2!} + \cdots + \frac{(K h)^{n}}{n!}\right] $$

Prove the sum converges as \(n\rightarrow\infty\)

The sum in part (b) is: $$ S_n \:=\: K h + \frac{(K h)^{2}}{2!} + \cdots + \frac{(K h)^{n}}{n!} $$ We need to prove that this sum converges as \(n \rightarrow \infty\). Observe that \(S_n\) is the partial sum of the series of the exponential function, i.e., \(\sum\limits_{n=0}^{\infty} \frac{(Kh)^n}{n!}\), which converges to \(e^{K h}\) for all \(K h \in \mathbb{R}\). Thus, the sum \(S_n\) converges as \(n\rightarrow\infty\). Since the sum in part (b) converges as \(n\rightarrow\infty\), it follows that the sum in part (a) is also convergent as \(n\rightarrow\infty\). This confirms that the sequence \(\left\\{\phi_{n}(t)\right\\}\) is convergent since it is the sequence of partial sums of a convergent infinite series.

Key concepts

Infinite Series Convergence

When we talk about infinite series convergence, we often encounter sequences of numbers whose terms are derived from some function or iterative procedure. In the context of differential equations, we typically deal with functions identified by sequences of approximations, which are expected to get closer to a solution as the number of terms increases.

Let's think of infinite series as a long train of numbers that are supposed to 'add up' to a particular value, even though there are infinitely many of them. Now, just because there are infinite numbers, it doesn't necessarily mean their sum is infinite. Some series have a finite sum, which is when we say the series is convergent. For instance, when a series represents a physical quantity, like a distance or a temperature change, if that series diverges, meaning the sum is not finite, it wouldn't make sense in the real world.

To establish whether the infinite series converges, we apply various convergence tests. In our exercise, we look at the series made from multiplying constant factors and a factorial, which is associated with the exponential function. If this series converges to a finite number, no matter how many terms we add, the sum doesn't change significantly after a point, showing that our function has a limit which is reachable, hence the function behaves 'nicely' and predictably. This is crucial for solving differential equations because it asserts that the approximation method used to find solutions will eventually stabilize, giving us a trustworthy answer. The concept of convergence is fundamental in calculus and analysis, and deeply connected with the behavior of functions in differential equations.

Triangle Inequality

The triangle inequality is a principle coming from geometry, which states that in any triangle, the length of one side is always less than the sum of the lengths of the other two sides. This concept might not seem directly related to sequences and series but is a powerful tool in calculus and analysis, especially dealing with sequences and their convergence.

When applied to mathematical expressions, particularly those involving absolute values, the triangle inequality tells us how to bound the magnitude of sums. It says that the absolute value of a sum of numbers is less than or equal to the sum of their absolute values. In the context of our differential equation, it helps us analyze the behavior of the approximation sequence by providing an upper bound on the terms we're adding together.

Understanding and being able to use the triangle inequality is vital in determining the convergence of a series because it allows us to replace complicated expressions with simpler ones that we know will be greater than or equal to the complexity we're trying to encapsulate. Therefore, it can often be the first step in proving that a series will converge as we did in step 1 of our exercise. This inequality is a tool that gives us a 'safe estimate' that ensures the true value lies within a certain range. This range gives us the control we need to analyze series and sequences safely.

Factorial

(!). The factorial of a positive integer, say \(n\), is the product of all positive integers less than or equal to \(n\). In mathematical terms, \(n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1\). It is a concept deeply ingrained in combinatorials, probability, and calculus.

In our infinite series, the factorial appears in the denominator. An intriguing trait of the factorial is that it grows extremely fast—much faster than exponential growth. This property is precisely why it is so useful in the convergence of series: it helps balance the series and prevent them from 'exploding' to infinity, thus aiding in making the series potentially convergent.

The factorial is not just some abstract idea—it has real-world applications. For instance, in physics, the Taylor series expansion uses factorials in the denominator, like in our differential equation problem. They are utilized to approximate the values of functions to any desired degree of accuracy. When working with series, especially those that can represent an exponential function as in our exercise (Step 3), the growth rate of factorials compared to the rest of the terms assures that the series will not diverge, which is why we can conclude the convergence of our sequence. The factorial plays a pivotal role in ensuring that complex functions have understandable and calculable properties.