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Chapter 9: Problem 4

Define finite sum and give an example.

Short Answer

Expert verified
Answer: A finite sum is the result of adding a specific and limited number of terms in a sequence or series. For example, consider an arithmetic sequence with the first term 2 and a common difference of 3. The sum of the first 5 terms of this sequence would be calculated as follows: 2 (the first term) + 5 (second term) + 8 (third term) + 11 (fourth term) + 14 (fifth term) = 40. Thus, the finite sum of the first 5 terms of this sequence is 40.

Step by step solution

01

Define Finite Sum

A finite sum is the summation of a limited number of terms in a sequence or a series. It is a specific type of sum where the number of terms being added is finite and known.
02

Simplify the Concept

In simpler terms, a finite sum is the result of adding a specific number of terms together. It can be represented mathematically as: \[\sum_{i=1}^{n} a_{i},\] where \(n\) is the number of terms in the sum and \(a_i\) represents the \(i\)-th term of the series.
03

Provide an Example

Let's consider an arithmetic sequence with the first term \(a_1 = 2\) and the common difference \(d = 3\). We want to find the sum of the first 5 terms of this sequence.
04

Determine the Terms in the Sequence

First, we need to determine the terms in the sequence: 1. The first term is \(a_1 = 2\). 2. The second term is \(a_2 = a_1 + d = 2 + 3 = 5\). 3. The third term is \(a_3 = a_2 + d = 5 + 3 = 8\). 4. The fourth term is \(a_4 = a_3 + d = 8 + 3 = 11\). 5. The fifth term is \(a_5 = a_4 + d = 11 + 3 = 14\). So, the terms in the sequence are: 2, 5, 8, 11, and 14.
05

Calculate the Finite Sum

Now we can calculate the finite sum of the first 5 terms: \[S = a_1 + a_2 + a_3 + a_4 + a_5 = 2 + 5 + 8 + 11 + 14 = 40\] The finite sum of the first 5 terms of this arithmetic sequence is 40.

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Most popular questions from this chapter

The famous Fibonacci sequence was proposed by Leonardo Pisano, also known as Fibonacci, in about A.D. 1200 as a model for the growth of rabbit populations. It is given by the recurrence relation \(f_{n+1}=f_{n}+f_{n-1}\), for \(n=1,2,3, \ldots,\) where \(f_{0}=0, f_{1}=1 .\) Each term of the sequence is the sum of its two predecessors. a. Write out the first ten terms of the sequence. b. Is the sequence bounded? c. Estimate or determine \(\varphi=\lim _{n \rightarrow \infty} \frac{f_{n+1}}{f_{n}},\) the ratio of the successive terms of the sequence. Provide evidence that \(\varphi=(1+\sqrt{5}) / 2,\) a number known as the golden mean. d. Verify the remarkable result that $$f_{n}=\frac{1}{\sqrt{5}}\left(\varphi^{n}-(-1)^{n} \varphi^{-n}\right)$$

Determine whether the following series converge or diverge. $$\sum_{k=2}^{\infty} \frac{4}{k \ln ^{2} k}$$

Evaluate the limit of the following sequences. $$a_{n}=\frac{4^{n}+5 n !}{n !+2^{n}}$$

Suppose a function \(f\) is defined by the geometric series \(f(x)=\sum_{k=0}^{\infty} x^{2 k}\) a. Evaluate \(f(0), f(0.2), f(0.5), f(1),\) and \(f(1.5),\) if possible. b. What is the domain of \(f ?\)

Explain the fallacy in the following argument. Let \(x=\frac{1}{1}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots\) and \(y=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\cdots .\) It follows that \(2 y=x+y\), which implies that \(x=y .\) On the other hand, $$x-y=\left(1-\frac{1}{2}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\left(\frac{1}{5}-\frac{1}{6}\right)+\cdots>0$$ is a sum of positive terms, so \(x>y .\) Thus, we have shown that \(x=y\) and \(x>y\).
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