Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

2.7.13 changed an equality to a inequality #10

Open
wants to merge 1 commit into
base: master
Choose a base branch
from
Open

2.7.13 changed an equality to a inequality #10

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
Binary file modified build/main.pdf
View file
Open in desktop
Binary file not shown.
6 changes: 3 additions & 3 deletions chapters/chapter2/chapter2-7.tex
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -230,8 +230,8 @@ \section{Properties of Infinite Series}
\begin{align*}
\abs{\sum_{j={m+1}}^n x_jy_j}
&= \abs{s_ny_{n+1} - s_my_{m+1} + \sum_{j=m+1}^n s_j(y_j - y_{j+1})}\\
&= \abs{s_ny_{n+1} - s_my_{m+1}} + \abs{\sum_{j=m+1}^n s_j(y_j - y_{j+1})} \tag{$\bigtriangleup$ inequality}\\
&= \abs{(s_n - s_m)y_{m+1}} + \abs{\sum_{j=m+1}^n s_j(y_j - y_{j+1})} \tag{$y_{m+1} > y_{n+1}$}\\
&\leq \abs{s_ny_{n+1} - s_my_{m+1}} + \abs{\sum_{j=m+1}^n s_j(y_j - y_{j+1})} \tag{$\bigtriangleup$ inequality}\\
&\leq \abs{(s_n - s_m)y_{m+1}} + \abs{\sum_{j=m+1}^n s_j(y_j - y_{j+1})} \tag{$y_{m+1} > y_{n+1}$}\\
&\leq M\abs{y_{m+1}} + M \abs{y_{m+1} - y_{n+1}}\\
&\leq 2M\abs{y_{m+1}}
\end{align*}
Expand Down Expand Up @@ -290,4 +290,4 @@ \section{Properties of Infinite Series}
then $A < \epsilon/(2b_1)$ and $\abs{\sum_{j={m+1}}^n x_jy_j} < 2\abs{b_1}\cdot \frac{\epsilon}{2\abs{b_1}}$,
which means we have the Cauchy Criterion for $\sum x_ny_n$, and therefore it converges.
}
}
}