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Essential Calculus: Early Transcendentals
Found in: Page 425
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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Short Answer

Explain what it means to say that\(\sum\limits_{n = 1}^\infty {{a_n}} = 5\).

It means that series converges to\(5\).

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Definition

A convergent series is an infinite sum of a sequence of numbers\(\sum\limits_{n = 1}^\infty {{a_n}} \) that adds up to finite number.

Explanation

Given a series \(\sum\limits_{n = 1}^\infty {{a_n}} = 5\)

It means that the sum of the infinite terms of the sequence \(\left\{ {{a_n}} \right\}\)is equal to 5.

i.e \({a_1} + {a_2} + {a_3} + \cdots \cdots = 5\)

We know that\(\mathop {\lim }\limits_{k \to \infty } {S_k} = \sum\limits_{n = 1}^\infty {{a_n}} \)where\({s_k}\)denotes the \({k^{{\rm{th }}}}\)partial sum of series.

Then, \(\mathop {\lim }\limits_{k \to \infty } {s_k} = 5\)and says that the series\(\sum\limits_{n = 1}^\infty {{a_n}} \)converges to 5.

And 5 is called the sum of the series\(\sum\limits_{n = 1}^\infty {{a_n}} \).

Most popular questions for Math Textbooks

Find the values of p for which the series is \(\sum\limits_{n = 1}^\infty {\frac{{lnn}}{{{n^p}}}} \)convergent.

After injection of a dose D of insulin, the concentration of insulin in a patient's system decays exponentially and so it can be written as \(D{e^{ - at}}\), where t represents time in hours and a is a positive constant.

(a) If a dose \(D\) is injected every \(T\) hours, write an expression for the sum of the residual concentrations just before the \((n + 1)\)st injection.

(b) Determine the limiting pre-injection concentration.

(c) If the concentration of insulin must always remain at or above a critical value \(C\), determine a minimal dosage \(D\) in terms of \(C\) , \(a\), and \(T\).

\(\sum\limits_{n = 2}^\infty {\frac{2}{{{n^2} - 1}}} \)

(a)Show that if \(\mathop {\lim }\limits_{n \to \infty } {a_2}_n = L\)and \(\mathop {\lim }\limits_{n \to \infty } {a_{2n + 1}} = L,\) then {\({a_n}\)} is convergent and \(\mathop {\lim }\limits_{n \to \infty } {a_n} = L\).

(a) If \({a_1} = 1\) and

\({a_{n + 1}} = 1 + \frac{1}{{1 + {a_n}}}\)

Find the first eight terms of the sequence {\({a_n}\)}. Then use part(a) to show that \(\mathop {\lim }\limits_{n \to \infty } {a_n} = \sqrt 2 \). This gives the continued fraction expansion

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Determine whether the geometric series is convergent or divergent..If it is convergent,find its sum.

\(\sum\limits_{n = 1}^\infty {\frac{{{{(10)}^n}}}{{{{( - 9)}^{n - 1}}}}} \)
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