Sums - Essential Dictionary II

Mathematical Writing - Vivaldi Franco 2014

Sums
Essential Dictionary II

Sums can be found in every corner of mathematics. We develop the dictionary and the very rich notation associated to summation.

Given a finite sequence of numbers $$(a_1,\ldots ,a_n)$$, we form the sum of its elements:

$$\begin{aligned} \sum _{k=1}^n a_k=a_1+a_2+\cdots +a_n. \end{aligned}$$

(3.5)

This notation was introduced by Fourier.1 It is called the (delimited) sigma-notation, as it makes use of the capital Greek letter with that name, called the summation symbol. The subscript $$k$$ is the index of summation, while $$1$$ and $$n$$ are, respectively, the lower limit and upper limit of summation. The quantity $$a_k$$ is the general term of the sum. The integer sequence $$(1,2,\ldots ,n)$$, specifying the values assumed by the index of summation, is called the range of summation.

The summation index is a dummy variable. This a variable used for internal book-keeping (like an integration variable), and its identity does not affect the value of the sum:

$$ \sum _{k=1}^n k^2=\sum _{j=1}^n j^2=1^2+2^2+\,\cdots \,+n^2. $$

I suggest that you adopt a summation symbol (one of $$i,j,k,l,m,n$$, see Sect. 6.2) and stick to it, unless there is a good reason to change it.

There are variants to the delimited sigma-notation (3.5). In an unrestricted sum, range information may be omitted altogether:

$$ \sum _k {n \atopwithdelims ()k}=2^n \quad n\geqslant 0. $$

(This sum has in fact only finitely many non-zero terms.) The summation range may also be specified by inequalities placed below the summation symbols:

$$\begin{aligned} \sum _{1\leqslant k\leqslant n}a_{k} \qquad \qquad \qquad \sum _{k\geqslant 1}a_{k} \qquad \qquad \qquad \sum _{1\leqslant j,k\leqslant N}a_{j,k}. \end{aligned}$$

(3.6)

Further conditions may be added to alter the range of summation:

$$\begin{aligned} \sum _{0<|k|\leqslant 2} a_k&= a_{-2}+a_{-1}+a_1+a_2\\ \sum _{1\leqslant k\leqslant 12\atop \mathrm{gcd}(k,12)=1} a_k&= a_1+a_5+a_7+a_{11}\\ \sum _{1\leqslant k\leqslant 10\atop k\,\, \text { prime}} a_k&= a_2+a_3+a_5+a_7. \end{aligned}$$

The advantage of this notation—called the standard form of the sigma-notation—is that the summation index is no longer restricted to a sequence of consecutive integers.

EXAMPLE. Consider the following manipulation:

$$\begin{aligned} \sum _{-2\leqslant k\leqslant n-3}2^{k+2} = \sum _{0\leqslant k+2\leqslant n-1}2^{k+2} = \sum _{0\leqslant j\leqslant n-1}2^j\, =\, 2^n-1. \end{aligned}$$

(3.7)

After adding $$2$$ to each term in the inequalities, we have simply replaced $$k+2$$ with $$j$$, obtaining the sum of a geometric progression which is evaluated explicitly. With this notation the change of summation index is unproblematic.

EXAMPLE. For any natural number $$n$$, let $$\varphi (n)$$ be the number of positive integers smaller than $$n$$ and relatively prime to it (with $$\varphi (1)=1$$). Thus $$\varphi (12)=4$$. This is Euler’s $${\varvec{\varphi }}$$ -function 2 of number theory, which is defined in symbols as follows:

$$\begin{aligned} \varphi (1)=1\qquad \qquad \varphi (n)= \sum _{1\leqslant k < n\atop \mathrm{gcd}(k,n)=1} 1,\qquad n>1. \end{aligned}$$

(3.8)

Simply by letting $$a_k=1$$ in (3.5), we have turned a summation into an algorithm that counts the elements of the set specified by the given conditions. This neat device illustrates the flexibility of the sigma-notation. Alternatively, Euler’s function may be defined as the cardinality of a set, using the cardinality symbol ’$$\#$$’ (see Sect. 2.1):

$$ \varphi (1)=1\qquad \qquad \qquad \varphi (n)= \#\{k\in \mathbb {N}\,:\, k< n,{\quad }\mathrm{gcd}(k,n)=1\} \qquad n>1. $$

Sums may be nested. A double sum is defined as follows:

$$\begin{aligned} \sum _{j=1}^J \sum _{k=1}^K a_{j,k}&\mathop {=}\limits ^\mathrm{def}\sum _{j=1}^J\left( \sum _{k=1}^K a_{j,k}\right) \\&=\sum _{k=1}^Ka_{1,k}+\sum _{k=1}^K a_{2,k} +\,\cdots \,+ \sum _{k=1}^K a_{J,k}. \end{aligned}$$

The sum in parentheses is a function of the outer summation index $$j$$; this sum is performed repeatedly, each time with a different value of $$j$$. The use of matching symbols for the index and the upper limit of summation ($$j,J,k,K$$) is particularly appropriate here. If the two ranges of summations are independent, inner and outer sums can be swapped. The commutative and associative laws of addition ensure that the value of the sum will not change. We illustrate this process with an example.

$$\begin{aligned} \sum _{j=0}^1\sum _{k=1}^{3} a_{j,k}&= (a_{0,1}+a_{0,2}+a_{0,3})+(a_{1,1}+a_{1,2}+a_{1,3})\\&= (a_{0,1}+a_{1,1})+(a_{0,2}+a_{1,2})+(a_{0,3}+a_{1,3})\\&= \sum _{k=1}^3\sum _{j=0}^{1} a_{j,k}. \end{aligned}$$

This chain of equalities adopts a standard layout and alignment. If the indices in a double sum have the same range, then they may be grouped together:

$$\begin{aligned} \sum _{i,j=1}^N a_{i,j}= \sum _{i=1}^N \sum _{j=1}^N a_{i,j}. \end{aligned}$$

(3.9)

A sum with infinitely many summands is called a series:

$$\begin{aligned} \sum _{k=1}^\infty a_k=a_1+a_2+\cdots . \end{aligned}$$

(3.10)

The summation range may be specified in several ways:

$$ \sum _{k\geqslant 1}a_k \qquad \sum _{k=-\infty }^\infty a_k \qquad \sum _{k\in \mathbb {Z}}a_k \qquad \sum _k a_k. $$

We haven’t yet explained what (3.10) means. We write the definition using the assignment operator:

$$\begin{aligned} \sum _{k=1}^\infty a_k \, \mathop {=}\limits ^{\nabla }\, \lim _{n\rightarrow \infty }\,\, \sum _{k=1}^na_k. \end{aligned}$$

(3.11)

The above limit—if it exists—is called the sum of the series, and the series is said to converge. Otherwise the series diverges. If a series has non-negative terms, then convergence is sometimes expressed with the suggestive notation (cf. (2.3), p. 11)

$$\begin{aligned} \sum _{k\geqslant 0}a_k<\infty . \end{aligned}$$

(3.12)

A power series is a series whose terms feature increasing powers of an indeterminate:

$$\begin{aligned} \sum _{k\geqslant 0} a_kx^k. \end{aligned}$$

(3.13)

For the values of $$x$$ for which the series (3.13) converges, the power series represents a function.

The terminology and notation introduced for sums extends with obvious modifications to products:

$$\begin{aligned} \prod _{k=1}^n a_k=a_1\cdot a_2\,\cdots \, a_n. \end{aligned}$$

(3.14)

An infinite product is called just that—there is no special name for it. Its value is defined as the limit of a sequence of finite products, as we did for sums (3.11). Infinite products are often written in the form

$$ \prod _{k\geqslant 0}(1+a_k) $$

because for convergence we must have $$a_k\rightarrow 0$$.

All constructs introduced above may be extended to combinations of sums and products.

Exercise 3.1

Write out the following sums in full.

$$\begin{aligned} \begin{array}{llll} 1. &{} \sum \limits _{0\leqslant k-1 < 3} a_k \qquad \qquad 2. &{} \sum \limits _{k^2 < 9} a_{-k} \qquad \qquad 3. &{} \sum \limits _{k^2\leqslant k+2} a_{1-k} \\ 4. &{} \sum \limits _{k\in 2\mathbb {Z}+1 \atop |k|<5} a_k \qquad \qquad \quad 5. &{} \sum \limits _{|k-3| < 5 \atop \gcd (k,6) > 1} a_k \qquad \quad 6. &{} \sum \limits _{k^2\leqslant 9} a_{k^2} \end{array} \end{aligned}$$