Question

Solve the following problems and discuss their relevance to the stopping criteria. a) Consider the sequence \(p_{n}\) where \(p_{n}=\sum_{i=1}^{n} \frac{1}{i} .\) Argue that \(p_{n}\) diverges, but \(\lim _{n \rightarrow \infty}\left(p_{n}-\right.\) \(\left.p_{n-1}\right)=0\) b) Let \(f(x)=x^{10}\). Clearly, \(p=0\) is a root of \(f,\) and the sequence \(p_{n}=\frac{1}{n}\) converges to \(p\). Show that \(f\left(p_{n}\right)<10^{-3}\) if \(n>1,\) but to obtain \(\left|p-p_{n}\right|<10^{-3}, n\) must be greater than \(10^{3}\).

Short answer

2. What is the limit of the difference between consecutive terms in the sequence as n approaches infinity? 3. Is \(f(p_n) < 10^{-3}\) if \(n > 1\) for the function \(f(x) = x^{10}\)? 4. To obtain \(|p - p_n| < 10^{-3}\), what should be the value of n?

Step-by-step solution

Part (a) - Proving \(p_n\) diverges

To show that the sequence \(p_n = \sum_{i=1}^n \frac{1}{i}\) diverges, we can consider the Harmonic series. The Harmonic series is well-known to diverge, and it is defined as: \(H_n = \sum_{i=1}^n \frac{1}{i}\) Since our sequence \(p_n\) is equal to the Harmonic series \(H_n\), we can conclude that \(p_n\) also diverges.

Part (a) - Limit of the difference between consecutive terms

Now we need to find the limit of the difference between consecutive terms in the sequence as n approaches infinity. Mathematically, we want to find: \(\lim_{n \rightarrow \infty} (p_n - p_{n-1})\) We can rewrite this as: \(\lim_{n \rightarrow \infty} \left(\sum_{i=1}^n \frac{1}{i} - \sum_{i=1}^{n-1} \frac{1}{i}\right)\) By canceling out the common terms, we get: \(\lim_{n \rightarrow \infty} \frac{1}{n}\) Since the limit of \(\frac{1}{n}\) as n approaches infinity is 0, we have: \(\lim_{n \rightarrow \infty} (p_n - p_{n-1}) = 0\)

Part (b) - Showing \(f(p_n) < 10^{-3}\) if \(n > 1\)

We have the function \(f(x) = x^{10}\) and the sequence \(p_n = \frac{1}{n}\). Let's calculate \(f(p_n)\) for \(n > 1\): \(f(p_n) = \left(\frac{1}{n}\right)^{10}\) Since \(n > 1\), \(\frac{1}{n} < 1\), which means that \(\left(\frac{1}{n}\right)^{10} < 10^{-3}\). Therefore, \(f(p_n) < 10^{-3}\) if \(n > 1\).

Part (b) - Finding the value of n for \(|p-p_n| < 10^{-3}\)

We know that \(p = 0\) is a root of the function \(f(x) = x^{10}\). We want to find the value of n for which \(|p - p_n| < 10^{-3}\). Since \(p = 0\), this translates to: \(|p_n| < 10^{-3}\) Recall that \(p_n = \frac{1}{n}\). Thus, we need to find the value of n such that: \(\left|\frac{1}{n}\right| < 10^{-3}\) Multiplying both sides by n and dividing both sides by \(10^{-3}\), we get: \(n > 10^{3}\) So, to obtain \(|p - p_n| < 10^{-3}\), n must be greater than \(10^{3}\).

Key concepts

Divergence of Sequences

In numerical analysis, the concept of divergence is used to describe a sequence that does not settle into a single value as it progresses to infinity. For a sequence to be divergent, the terms must either increase or decrease without bound, or oscillate without approaching a specific value. Understanding divergence is crucial when analyzing algorithms or series in mathematical problems.

An illustrative example of a divergent sequence is the Harmonic series. Despite adding infinitesimally smaller terms with each step, this series never converges to a fixed value. This behavior is counterintuitive since one might expect that continually adding smaller values would eventually stop making a significant difference. A key takeaway is that divergence does not imply that the difference between consecutive terms doesn't approach zero, rather that the sum as a whole doesn't converge.

• A sequence \(a_n\) is said to diverge to infinity if for every positive real number M, there exists a natural number N such that for all n \geq N, \(a_n \geq M\).
• A sequence oscillates if it does not approach a particular value as n increases.

Harmonic Series

The Harmonic series is a famous example in the study of infinite series and sequences. It is defined as the infinite sum of reciprocals of natural numbers, expressed formally as \(H_n = \sum_{i=1}^n \frac{1}{i}\). While each term gets smaller and smaller, the overall sum grows without bound, thus the series diverges.

Here's a curious aspect of the Harmonic series: as you take the difference between the nth term of the series and the (n-1)th term, which is simply \(\frac{1}{n}\), this difference approaches zero as n grows large. This fact sometimes gives the false impression that the series might be converging. However, when you examine the sum of all terms up to the nth term, it grows indefinitely. This is a foundational example in understanding the subtle nature of infinite series and why precise definitions and proofs are necessary in mathematics.

• The Harmonic series highlights the difference between individual term behavior and the behavior of the sum as a whole.
• Its divergence is often proved using the comparison test or the integral test in Calculus.

Convergence Criteria

Convergence criteria are the rules or tests used to determine whether a sequence or series converges to a limit. In the context of series, convergence means that adding the infinite number of terms yields a finite sum, while for sequences, it implies the terms approach a specific value as the sequence progresses. Various tests can be applied to assess convergence, each with its own conditions and utility.

Some common convergence tests include:

• The Nth Term Test: If the limit of the nth term of a sequence or series does not approach zero, the series does not converge.
• The Ratio Test: If the absolute ratio of consecutive terms approaches a limit less than 1, the series converges.
• The Comparison Test: Comparing a series to another well-known series can help determine convergence or divergence.
• The Integral Test: This test connects the sum of a series to the area under a curve, utilizing the properties of integrals to determine convergence.

It's important to note that no single test is universally applicable, so it's essential to understand the context and characteristics of the sequence or series under consideration.

Roots of Functions

The concept of the root of a function is foundational in mathematics, particularly in algebra and calculus. A root of a function f(x) is a number \(x = p\) such that \(f(p) = 0\). Finding the roots of a function is often associated with solving equations, and the roots represent the x-values where the graph of the function crosses the x-axis.

• Zeroes of polynomials: These are the x-values for which the polynomial equals zero. For example, if \(f(x) = x^2 - 4\), the roots are x = -2 and x = 2.
• Analytical methods: Include solving equations algebraically, such as factoring or using the quadratic formula.
• Numerical methods: Involve iterative algorithms, such as Newton-Raphson or bisection method, to approximate roots when analytical solutions are not feasible.

Numerical analysis often deals with iterative methods for approximating roots, where successive approximations converge to a root. Understanding the precision of these approximations is critical, as the exercise shows. It demonstrates that even a small value for \(p_n\) could yield an f-value within acceptable error bounds, yet the approximation to the root itself may require a much finer approach.