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Chapter 3: Problem 90

For the sequence a defined by \(a_{n}=\frac{n-1}{n^{2}(n-2)^{2}}, \quad n \geq 3\) and the sequence \(z\) defined by \(z_{n}=\sum_{i=3}^{n} a_{i}\). Find \(z_{3}\)

Short Answer

Expert verified
The value of \(z_3\) is \(\frac{2}{9}\)

Step by step solution

01

Find the Value of \(a_3\)

Firstly, substitute 'n = 3' into the given formula for 'a', which is \(a_n = \frac{n-1}{n^{2}(n-2)^{2}}\). Thus, \(a_3 = \frac{3-1}{3^{2}(3-2)^{2}} = \frac{2}{9}\).
02

Calculate \(z_3\)

Substitute 'n = 3' into the definition of the sequence 'z'. Given that \(z_n = \sum_{i=3}^{n} a_{i}\), \(z_3 = a_3 = \frac{2}{9}\)

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Key Concepts

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

Discrete Mathematics
Discrete mathematics is a branch of mathematics that deals with discreet, typically countable and non-continuous structures. Unlike calculus, which handles continuous variables and change, discrete mathematics covers topics such as logic, set theory, graph theory, and combinatorics. It's essential in computer science, encryption, and network design.

When students consider sequences and series, they are delving into one of the most fundamental aspects of discrete mathematics. Sequences are ordered lists of numbers that follow a specific rule or pattern, while series are the sum of sequences' elements. Understanding the structure of sequences and how to sum their components is critical for solving numerous problems in this field.
Infinite Series
An infinite series is a sum of the terms of an infinite sequence. It's an expression of the form \( a_1+a_2+a_3+... \), where \( a_i \) represents the \(i\)-th term in the sequence. However, not all infinite series converge to a finite value. In mathematics, we seek to understand when and how these series reach a finite limit, called their sum.

To determine convergence, various tests like the Ratio Test, Root Test, and others can be applied. These series are pivotal in mathematical analysis and theoretical concepts, as well as practical applications such as calculating interests in finance or in the behavior of electrical circuits.
Mathematical Sequences
A mathematical sequence is a collection of objects or numbers arranged in a particular order. In the context of the exercise provided, the sequence \( a_n \) is defined by a function that dictates the value of each term based on its position, \( n \). To analyze such sequences, one might consider the behavior of its terms as \( n \) grows larger鈥攄oes the sequence approach a limit, oscillate, or perhaps grow without bound?

Sequences are also the groundwork for series. As in the example with sequence \( z \), a series is formed by summing the terms of a sequence over a range of indices. The calculation of the series' terms often involves methods from calculus, should the terms' formula be complex, or simple arithmetic for more straightforward cases. In the exercise, once \( a_3 \) is found, it also represents the value of \( z_3 \) since the summation starts and ends at 3, making \( z_3 \) a summation of only one term.

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Most popular questions from this chapter

For the sequence z defined by $$z_{n}=(2+n) 3^{n}, \quad n \geq 0$$. Prove that \(\left\\{z_{n}\right\\}\) satisfies $$ z_{n}=6 z_{n-1}-9 z_{n-2}, \quad n \geq 2 $$

In give an example of a unary operator \([\) different from \(f(x)=x\), for all \(x\) ] on the given set. $$ \text { The set of all finite subsets of }\\{1,2,3, \ldots\\} $$

Let \(X\) be the set of positive integers that are not perfect squares. (A perfect square \(m\) is an integer of the form \(m=i^{2}\) where \(i\) is an integer.) Concern the sequence s from \(X\) to \(\mathbf{Z}\) defined as follows. If \(n \in X,\) let \(s_{n}\) be the least integer \(a_{k}\) for which there exist integers \(a_{1}, \ldots, a_{k}\) with \(n

Let \(L\) be the set of all strings, including the null string, that can be constructed by repeated application of the following rules: If \(\alpha \in L,\) then \(a \alpha b \in L\) and \(b \alpha a \in L .\) If \(\alpha \in L\) and \(\beta \in L,\) then \(\alpha \beta \in L\) For example, \(a b\) is in \(L\), for if we take \(\alpha=\lambda,\) then \(\alpha \in L\) and the first rule states that \(a b=a \alpha b \in L\). Similarly, \(b a \in L\). As another example, aabb is in \(L\), for if we take \(\alpha=a b\), then \(\alpha \in L\); by the first rule, \(a a b b=a \alpha b \in L .\) As a final example, aabbba is in \(L\), for if we take \(\alpha=a a b b\) and \(\beta=b a,\) then \(\alpha \in L\) and \(\beta \in L ;\) by the second rule, \(a a b b b a=\alpha \beta \in L\). Prove that if \(\alpha\) has equal numbers of \(a\) 's and \(b\) 's, then \(\alpha \in L\).

Use the following definitions. Let \(U\) be a universal set and let \(X \subseteq U\). Define $$ C_{X}(x)=\left\\{\begin{array}{ll} 1 & \text { if } x \in X \\ 0 & \text { if } x \notin X . \end{array}\right. $$ We call \(C_{X}\) the characteristic function of \(X(\) in \(U) .\) (A look ahead at the next Problem-Solving Corner may help in understanding the following exercises.) Prove that \(C_{\bar{X}}(x)=1-C_{X}(x)\) for all \(x \in U\).
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