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Chapter 8: Problem 2

What is the difference between a geometric sum and a geometric series?

Short Answer

Expert verified
Answer: A geometric series is an infinite sequence of terms with a constant ratio between them, while a geometric sum is the calculated value obtained by adding the terms of the geometric series up to a specified index n. A geometric sum is expressed as a formula, while a geometric series is an ordered list of terms.

Step by step solution

01

Define a geometric series

A geometric series is an infinite sequence of terms in which each term is the product of the previous term and a constant (common ratio, often denoted as "r"). In other words, a geometric series is formed when successive terms in the sequence are multiplied by the same constant. The general form of a geometric series can be written as: S_n = a_1 + a_2 + a_3 + ... + a_n = a_1 (1 + r + r^2 + ... + r^(n-1)) Here, S_n represents the sum of the first n terms in the series, a_1 is the first term, r is the common ratio between terms, and n is the total number of terms.
02

Define a geometric sum

A geometric sum is the result of adding up all the terms in a geometric series up to a specified index n. Since a geometric series is an ordered sequence of terms, a geometric sum is the value calculated by adding all those terms together up to the nth term. The formula for finding the sum of the first n terms of a geometric series (geometric sum) is: S_n = a_1 (1 - r^n) / (1 - r) Where S_n is the geometric sum of the first n terms, a_1 is the first term, r is the common ratio, and n is the total number of terms.
03

Highlight the differences

The main differences between a geometric sum and a geometric series are: 1. A geometric series is an infinite sequence of terms with a constant (common) ratio between them, while a geometric sum is the calculated value obtained by adding the terms of the geometric series up to a specified index n. 2. A geometric sum is expressed as a formula, which is a function of the first term (a_1), the common ratio (r), and the number of terms (n), whereas a geometric series is expressed as an ordered list of terms. In summary, a geometric series represents the sequence of terms with a constant ratio, while a geometric sum is the calculated value resulting from adding those terms together up to a certain point in the series.

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Most popular questions from this chapter

Use Theorem 8.6 to find the limit of the following sequences or state that they diverge. $$\left\\{\frac{n^{10}}{\ln ^{20} n}\right\\}$$

Find a formula for the nth term of the sequence of partial sums \(\left\\{S_{n}\right\\} .\) Then evaluate lim \(S_{n}\) to obtain the value of the series or state that the series diverges.\(^{n \rightarrow \infty}\) $$\sum_{k=1}^{\infty}\left(\tan ^{-1}(k+1)-\tan ^{-1} k\right)$$

A fishery manager knows that her fish population naturally increases at a rate of \(1.5 \%\) per month. At the end of each month, 120 fish are harvested. Let \(F_{n}\) be the fish population after the \(n\) th month, where \(F_{0}=4000\) fish. Assume that this process continues indefinitely. Use infinite series to find the longterm (steady-state) population of the fish.

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\frac{n^{8}+n^{7}}{n^{7}+n^{8} \ln n}$$

Pick two positive numbers \(a_{0}\) and \(b_{0}\) with \(a_{0}>b_{0},\) and write out the first few terms of the two sequences \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}:\) $$a_{n+1}=\frac{a_{n}+b_{n}}{2}, \quad b_{n+1}=\sqrt{a_{n} b_{n}}, \quad \text { for } n=0,1,2 \dots$$ (Recall that the arithmetic mean \(A=(p+q) / 2\) and the geometric mean \(G=\sqrt{p q}\) of two positive numbers \(p\) and \(q\) satisfy \(A \geq G.)\) a. Show that \(a_{n} > b_{n}\) for all \(n\). b. Show that \(\left\\{a_{n}\right\\}\) is a decreasing sequence and \(\left\\{b_{n}\right\\}\) is an increasing sequence. c. Conclude that \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) converge. d. Show that \(a_{n+1}-b_{n+1}<\left(a_{n}-b_{n}\right) / 2\) and conclude that \(\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n} .\) The common value of these limits is called the arithmetic-geometric mean of \(a_{0}\) and \(b_{0},\) denoted \(\mathrm{AGM}\left(a_{0}, b_{0}\right)\). e. Estimate AGM(12,20). Estimate Gauss' constant \(1 / \mathrm{AGM}(1, \sqrt{2})\).
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