deletions | additions
diff --git a/alternating series.tex b/alternating series.tex
index f3d73c2..132aa3f 100644
--- a/alternating series.tex
+++ b/alternating series.tex
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Ratio Test:
\begin{align}
L\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left|b(n\:+\: 1)\hspace{1em}\div\hspace{1em}b(n)\right|
\tag{15}\\[1em] \\[1em]
&=\hspace{1em}\lim_{n\to\infty} \left|\frac{(n\:+\:1\:+\:1)}{(n\:+\:1)^3}\hspace{1em}\div\hspace{1em}\frac{(n\:+\:1)}{n^3}\right|
\tag{16}\\[1em] \\[1em]
&=\hspace{1em}\lim_{n\to\infty} \left|\frac{(n\:+\:2)}{(n\:+\:1)^3}\hspace{1em}\div\hspace{1em}\frac{(n\:+\:1)}{n^3}\right|
\tag{17}\\[1em] \\[1em]
&=\hspace{1em}\lim_{n\to\infty} \left|\frac{(n\:+\:2)}{(n\:+\:1)^3}\hspace{1em}\div\hspace{1em}\frac{(n\:+\:1)}{n^3}\right|
\tag{18}\\[1em] \\[1em]
&=\hspace{1em}\lim_{n\to\infty} \left|\frac{(n\:+\:2)}{(n\:+\:1)^3}\hspace{1em}\cdot\hspace{1em}\frac{n^3}{(n\:+\:1)}\right|
\tag{19}\\[1em] \\[1em]
&=\hspace{1em}\lim_{n\to\infty} \left|\frac{n^3(n\:+\:2)}{(n\:+\:1)(n\:+\:1)^3}\right|
\tag{20}\\[1em] \\[1em]
&=\hspace{1em}\lim_{n\to\infty} \left|\frac{n^3(n\:+\:2)}{(n\:+\:1)(n\:+\:1)(n\:+\:1)(n\:+\:1)}\right|
\tag{21}\\[1em] \\[1em]
&=\hspace{1em}\lim_{n\to\infty} \left|\frac{n^3(n\:+\:2)}{(n^2\:+\:2n\:+\:1)(n\:+\:1)(n\:+\:1)}\right|
\tag{22}\\[1em] \\[1em]
&=\hspace{1em}\lim_{n\to\infty} \left|\frac{n^3(n\:+\:2)}{(n^3\:+\:3n^2\:+\:3n\:+\:1)(n\:+\:1)}\right|
\tag{23}\\[1em] \\[1em]
&=\hspace{1em}\lim_{n\to\infty} \left|\frac{n^3(n\:+\:2)}{(n^4\:+\:4n^3\:+\:6n^2\:+\:4n\:+\:1)}\right|
\tag{24}\\[1em] \\[1em]
&=\hspace{1em}\lim_{n\to\infty} \left|\frac{n^4\left(\frac{n}{n}\:+\:\frac{2}{n}\right)}{n^4\left(\frac{n^4}{n^4}\:+\:\frac{4n^3}{n^4}\:+\:\frac{6n^2}{n^4}\:+\:\frac{4n}{n^4}\:+\:\frac{1}{n^4}\right)}\right|
\tag{25}\\[1em] \\[1em]
&=\hspace{1em}\lim_{n\to\infty} \left|\frac{\left(1\:+\:\frac{2}{n}\right)}{\left(1\:+\:\frac{4}{n}\:+\:\frac{6}{n^2}\:+\:\frac{4}{n^3}\:+\:\frac{1}{n^4}\right)}\right|
\tag{26}\\[1em] \\[1em]
&=\hspace{1em} \left|\frac{\left(1\:+\:\frac{2}{\infty}\right)}{\left(1\:+\:\frac{4}{\infty}\:+\:\frac{6}{\infty}\:+\:\frac{4}{\infty}\:+\:\frac{1}{\infty}\right)}\right|
\tag{27}\\[1em] \\[1em]
&=\hspace{1em} \left|\frac{\left(1\:+\:0\right)}{\left(1\:+\:0\:+\:0\:+\:0\:+\:0\right)}\right|
\tag{28}\\[1em] \\[1em]
&=\hspace{1em} \left|\frac{1}{1}\right|
\tag{28}\\[1em] \\[1em]
L\hspace{1em}&=\hspace{1em} [1]
\tag{29}\\[1em] \\[1em]
\end{align}
Ratio Test Condidtions:
...
Raabe's Test:
\begin{align}
L\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left[n\cdot\left(b(n)\hspace{1em}\div\hspace{1em}b(n\:+\:1)\right)\hspace{1em}-\hspace{1em}1\right]
\tag{30}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left[n\cdot\left(\frac{(n\:+\:1)}{n^3}\hspace{1em}\div\hspace{1em}\frac{(n\:+\:1\:+\:1)}{(n\:+\:1)^3}\right)\hspace{1em}-\hspace{1em}1\right]
\tag{31}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left[n\cdot\left(\frac{(n\:+\:1)}{n^3}\hspace{1em}\div\hspace{1em}\frac{(n\:+\:2)}{(n\:+\:1)^3}\right)\hspace{1em}-\hspace{1em}1\right]
\tag{32}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left[n\cdot\left(\frac{(n\:+\:2)}{n^3}\hspace{1em}\cdot\hspace{1em}\frac{(n\:+\:1)^3}{(n\:+\:1)}\right)\hspace{1em}-\hspace{1em}1\right]
\tag{33}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left[n\cdot\left(\frac{(n\:+\:1)(n\:+\:1)^3}{n^3(n\:+\:2)}\right)\hspace{1em}-\hspace{1em}1\right]
\tag{34}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left[n\cdot\left(\frac{(n\:+\:1)(n\:+\:1)(n\:+\:1)(n\:+\:1)}{n^3(n\:+\:2)}\right)\hspace{1em}-\hspace{1em}1\right]
\tag{35}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left[n\cdot\left(\frac{(n^2\:+\:2n\:+\:1)(n\:+\:1)(n\:+\:1)}{n^3(n\:+\:2)}\right)\hspace{1em}-\hspace{1em}1\right]
\tag{36}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left[n\cdot\left(\frac{(n^3\:+\:3n^2\:+\:3n\:+\:1)(n\:+\:1)}{n^3(n\:+\:2)}\right)\hspace{1em}-\hspace{1em}1\right]
\tag{37}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left[n\cdot\left(\frac{(n^4\:+\:4n^3\:+\:6n^2\:+\:4n\:+\:1)}{n^3(n\:+\:2)}\right)\hspace{1em}-\hspace{1em}1\right]
\tag{38}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left[n\cdot\left(\frac{(n^4\:+\:4n^3\:+\:6n^2\:+\:4n\:+\:1)}{(n^4\:+\:2n^3)}\right)\hspace{1em}-\hspace{1em}1\right]
\tag{39}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left[n\cdot\left(\frac{(n^4\:+\:4n^3\:+\:6n^2\:+\:4n\:+\:1)}{(n^4\:+\:2n^3)}\right)\hspace{1em}-\hspace{1em}\frac{(n^4\:+\:2n^3)}{(n^4\:+\:2n^3)}\right]
\tag{40}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left[n\cdot\left(\frac{(n^4\:+\:4n^3\:+\:6n^2\:+\:4n\:+\:1)\hspace{1em}-\hspace{1em}(n^4\:+\:2n^3)}{(n^4\:+\:2n^3)}\right)\right]
\tag{41}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left[n\cdot\left(\frac{(n^4\:+\:4n^3\:+\:6n^2\:+\:4n\:+\:1\:-\:n^4\:-\:2n^3)}{(n^4\:+\:2n^3)}\right)\right]
\tag{42}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left[n\cdot\left(\frac{(2n^3\:+\:6n^2\:+\:4n\:+\:1)}{(n^4\:+\:2n^3)}\right)\right]
\tag{43}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left[\frac{n\cdot(2n^3\:+\:6n^2\:+\:4n\:+\:1)}{(n^4\:+\:2n^3)}\right]
\tag{44}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left[\frac{n\cdot n^3\cdot\left(\frac{2n^3}{n^3}\:+\:\frac{6n^2}{n^3}\:+\:\frac{4n}{n^3}\:+\:\frac{2}{n^3}\right)}{n^4\cdot\left(\frac{n^4}{n^4}\:+\:\frac{n^3}{n^4}\right)}\right]
\tag{45}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em}\lim_{n\to\infty} \left[\frac{\left(2\:+\:\frac{6}{n}\:+\:\frac{4}{n^2}\:+\:\frac{1}{n^3}\right)}{\left(1\:+\:\frac{1}{n}\right)}\right]
\tag{46}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em} \left[\frac{\left(2\:+\:\frac{6}{\infty}\:+\:\frac{4}{\infty}\:+\:\frac{1}{\infty}\right)}{\left(1\:+\:\frac{1}{\infty}\right)}\right]
\tag{47}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em} \left[\frac{\left(2\:+\:0\:+\:0\:+\:0\right)}{\left(1\:+\:0\right)}\right]
\tag{48}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em} \left[\frac{\left(2\right)}{\left(1\right)}\right]
\tag{49}\\[1em] \\[1em]
\hspace{1em}&=\hspace{1em} \left[2\right]
\tag{50}\\[1em] \\[1em]
\end{align}
Raabe's Conditions:
