Guide to LaTeX (4th Edition) (Tools and Techniques

12.2. Standard features of A M S-L AT E X 273
\begin{equation}\begin{split}
H_c&=\frac{1}{2n}\sum_{l=0}ˆn (-1)ˆl (k-l)ˆ{p-2}
\sum_{l_1+\dots+l_p=l} \prod_{i=1}ˆp \binom{n_i}{l_i}\\
&\quad\times[(k-l) - (k_i-l_i)]ˆ{k_i-l_i}\times
\Bigl[(k-l)ˆ2 - \sum_{j=1}ˆp (k_i-l_i)ˆ2\Bigr]
\end{split}\end{equation}
The alignment has been chosen to be just before the equals sign. The
second line begins with the alignment marker & so that the equation
continues below the = in the first line. A \quad has been inserted so
that the × is not immediately below the equals sign. Alternatively, one
could place the & after the = and dispense with \quad in the second line.
However, the above is more suitable when there are continuation lines
beginning with =. Note the centered equation number at the right.
The gather environment
The gather environment switches to math mode, centering each of its
formula lines without any alignment. The formula lines are separated by
\\ commands. Each line receives an equation number, unless the *-form
has been used, or \notag has been issued in that line.
1
2 +
4 9 n2 ∞ n2 2 3
n
n
+ + · · · +
+ · · · =
(12.3)
3 4
n + 1
n + 1
n=1
⎛
n2 n n
⎜
converges since lim
= lim ⎜ 1
n→∞ n + 1 n→∞ ⎝
1 + 1
⎞n
⎟
⎠ =
n
1
< 1
e
root condition
2 + 3
∞
4 n + 1
n + 1
+ + · · · + + · · · =
4 9 n2 n
n=1
2
(12.4)
∞
x + 1
diverges since
dx = ln x −
x2 1
∞ = ∞ (integral condition)
x
c
\begin{gather}
\frac{1}{2} + \left(\frac{2}{3}\right)ˆ4 + \left(\frac{3}{4}
\right)ˆ9 + \dots + \left(\frac{n}{n+1}\right)ˆ{nˆ2} + \dotsb
= \sum_{n=1}ˆ\infty \left(\frac{n}{n+1}\right)ˆ{nˆ2} \\
\text{converges since}\quad\lim_{n\to\infty}
\sqrt[n]{\left(\frac{n}{n+1}\right)ˆ{nˆ2}} = \lim_{n\to\infty}
\left(\frac{1}{1 + \dfrac{1}{n}}\right)ˆn = \frac{1}{\me} < 1
\tag*{root condition}\\
2 + \frac{3}{4} + \frac{4}{9} + \dots + \frac{n+1}{nˆ2} +
\dotsb = \sum_{n=1}ˆ\infty \frac{n+1}{nˆ2}\\
\text{diverges since}\quad\int_cˆ\infty \frac{x+1}{xˆ2}\,
c