Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 11 - Infinite Sequences and Series - 11.2 Exercises - Page 735: 16

Answer

a) The two sums are the same in what they represent, but they differ in indices. b) The two sums are different.

Work Step by Step

a) The two sums are the same in what they represent, but they 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 the 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 $n$ is equal to $n*a_{j}$ .
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