The commonest hand-written style for expressions is to place the limit
expressions on operators such as `\sum`

and `\int`

physically
above and below the operator. In (La)TeX, we write these limit
expressions using sub- and superscripts applied to the operator, but
they don’t always appear in the “handwritten” way in TeX’s
output.

The reason is, that when an expression appears in non-display maths,
in running text (and is therefore in TeX `\textstyle`

), placing
the limits thus could lead to ragged line spacing (and hence
difficult-to-read text). It is therefore common (in `\textstyle`

)
to place the limits as one would sub- and superscripts of variables.

This is not universally satisfactory, so the primitive `\limits`

is
provided:

which will place the limits right above and below the symbol (and be blowed to the typography…).$\sum\limits_{n=1}^{m} ...$

Contrariwise, you may wish to change the arrangement of the limits
when in `\displaystyle`

. For this purpose, there’s a corresponding
`\nolimits`

:

which will place the limits as they would be in\[\sum\nolimits_{n=1}^{m} ...\]

`\textstyle`

.
Alternatively, one can manipulate the
`\textstyle`

/`\displaystyle`

state of the mathematics. To get
“`\limits`

placement” in inline maths,

and for “$\displaystyle\sum_{n=1}^{m} ...$

`\nolimits`

placement” in display maths,
`\nolimits`

:
will serve. Either of these forms may have effects other than on the operator you’re considering, but there are still those who prefer this formulation.\[\textstyle\sum_{n=1}^{m} ...\]

Remember, if you’re declaring a special operator of your own, the AMSLaTeX functions (that you ought to be using) allow you to choose how limits are displayed, at definition time.

(Note that the macro `\int`

normally has `\nolimits`

built in to
its definition. There is an example in the TeXbook to show how odd
`\int`

`\limits`

looks when typeset.)

This question on the Web: http://www.tex.ac.uk/cgi-bin/texfaq2html?label=limits