Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 11 - Infinite Sequences and Series - 11.2 Series - 11.2 Exercises - Page 756: 16


a) There is no difference. The two notations are identical in what they represent but only differ in indices. They both represent a summation of $n$ terms of the sequence $a_{i}$ or $a_{j}$ which looks like the following: $a_{1}$+$a_{2}$+$a_{3}$ +...+$a_{n}$ = $\Sigma a_{x}$, here $x$ is either $i$ or $j$ . b) The first notation means same as part a, being the following: $a_{1}$+$a_{2}$+$a_{3}$ +...+$a_{n}$ = $\Sigma a_{i}$ , that is, the sum of all terms of $a_{i}$ which is influenced by the index i progressing from $1$ to $n$. On the other hand, the notation where the term is $a_{j}$ but the index of the summation symbol is $i$ starting from $1$ going to $n$ , that is the second summation equals to $n*a_{i}$ .

Work Step by Step

a) There is no difference in the two summation in 16.(a) because the change of indexes from $i$ to $j$ doesn't change what both notations represent, that is the following summation to the nth term: $x_{1}$+$x_{2}$+$x_{3}$ +...+$x_{n}$ b) The first notation means $x_{1}$+$x_{2}$+$x_{3}$ +...+$x_{n}$ thus $i$ goes from 1 to $n$ ; however the other summation that has an index in the of j ($a_{j}$) but the index of the summation $\Sigma$ is $i$ from 1 to $n$ , which means that the second summation is $n$ times the value of ($a_{j})$ = n*($a_{j}$).
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