Sums, Integrals and Limits in LaTeX: \sum, \int, \lim

The big operators: sums, products, integrals and limits, and where their limits go. About ten minutes.

Sums, products, integrals and limits are the "big operators" of maths. Each is one command, and each takes its limits from _ and ^, the same two characters as subscripts and powers.

Sums

\sum gives the summation sign, with the bottom limit in _ and the top in ^:

\sum_{i=1}^{n} i = \frac{n(n+1)}{2}

Edit the LaTeX

In a formula on its own line, like this one, the limits sit under and over the sign. Inside a sentence they move to the side to save height: $\sum_{i=1}^{n} i$ . That is LaTeX's choice, and usually the right one.

Products

\prod works just like \sum:

\prod_{k=1}^{n} k = n!

Edit the LaTeX

Bigger versions of the set symbols work the same way too: \bigcup_{i=1}^{n} A_i gives $\bigcup_{i=1}^{n} A_i$ .

Integrals

\int gives the integral sign. Its limits also come from _ and ^, but they stay at the side of the sign, as in printed books:

\int_a^b f(x) \, dx = F(b) - F(a)

Edit the LaTeX

The \, before the dx adds a thin space between the function and the dx, which is how integrals are usually printed. Limits longer than one character need braces: \int_{-\infty}^{\infty}.

There are signs for multiple integrals and for an integral round a closed curve: \iint gives $\iint$ , \iiint gives $\iiint$ , and \oint gives $\oint$ .

\iint_R f(x, y) \, dA

Edit the LaTeX

Limits

\lim prints "lim" upright, with what tends to what in _ underneath:

\lim_{x \to 0} \frac{\sin x}{x} = 1

Edit the LaTeX

\to gives the arrow and \infty infinity: \lim_{n \to \infty}. As with sums, the condition sits underneath on its own line and at the side in a sentence.

Practise

$\sum_{k=1}^{n} k^2$
$\prod_{k=1}^{n} k = n!$
$\int_0^{\pi} \sin x \, dx = 2$
$\int_a^b f(x) \, dx = F(b) - F(a)$
$\lim_{x \to \infty} \frac{1}{x} = 0$
$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$
$\iint_R xy \, dA$
$\sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x$

In short

\sum, \prod and \int take their limits from _ and ^.
Sum and product limits sit under and over in a displayed formula, and at the side in a sentence; integral limits stay at the side.
\lim_{x \to a} writes a limit; \, before dx adds the usual thin space.

Next, in lesson 7: function names like sin and log, and derivatives.

Sums, integrals and limits

Sums

Products

Integrals

Limits

Practise

In short

Would you rather write it by hand?