The summation symbol \(\sum\)

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The summation symbol is simply a shorthand notation for a sum of many numbers or variables. Let's see some examples first.

Example. The sum 

\[1 + 2 + 3 + \cdots + 10\]

can be written as

\[\sum_{i=1}^{10}i.\]

That is,

\[\sum_{i=1}^{10}i = 1 + 2 + 3 + \cdots + 10.\]

Here, the symbol \(\sum\) is called the summation symbol, \(i\) is the index. The above expression means we take the sum of \(i\) where the value of \(i\) varies from 1 to 10. □

Remark. The letter \(\sum\) (pronounced sigma) is an uppercase letter from the Greek alphabet that corresponds to the letter S in the English alphabet. Why S? Because \(\Sigma\) stands for Summation. □

Let's see some more examples.

Example.  

\[\sum_{k=1}^{10} k^2 = 1^2 + 2^2 + 3^2 + \cdots + 10^2.\]

We can use any letter for the index (\(i\), \(k\), etc.). □

Example. 

\[\sum_{k=1}^{100} 1 = 1 + 1 + \cdots + 1 \text{(adding 1 for100 times)}\]

The index needs not appear in the ``body'' of the summation symbol. In this case, the body is the constant 1. □

Example.

\[\sum_{k=-5}^{5}(2k^2 + 1) = (2\times (-5)^2 + 1) + (2\times (-4)^2 + 1) + \cdots + (2\times 4^2 + 1) + (2\times 5^2 + 1).\]

The index can vary between arbitrary values. In this case, it varies from -5 to +5. □

Now, let's consider a sequence of numbers \(a_1, a_2, \cdots, a_n, \cdots\). The sum of the first \(n\) terms of this sequence can be written as

\[\sum_{i=1}^{n} a_i = a_1 + a_2 + \cdots + a_n.\]

As noted above, the letter used for the index does not matter. Therefore, this sum is the same as

\[\sum_{j=1}^{n} a_j\]

or 

\[\sum_{k=1}^{n} a_k\]

or

\[\sum_{\alpha=1}^{n} a_\alpha,\]

etc. However, if a symbol is used as a variable outside the summation symbol, you should not use the same symbol as the index of the sum.

Example.

Let \(k = 10\), that is, \(k\) is a variable representing the constant 10. To express \(k\times (1 + 2 + \cdots + 10)\), we should not write

\[k\sum_{k=1}^{10}k\]

because the first factor \(k\) is different from the index \(k\) of the sum. Instead, we should write something like

\[k\sum_{j=1}^{10}j.\]

This way, we can avoid confusion. If a variable is defined outside a summation symbol, that variable is called a free variable (with respect to the sum). On the other hand, the variable used as the index in a sum is called a bound variable because its context (i.e. the extent to which the variable is meaningful) is bound in the sum. □

When the range of the index is not important or obvious from the context, we may omit the upper and lower bounds to write simply

\[\sum_i a_i.\]

Some properties of the summation symbol.

As noted above, the summation symbol is just a shorthand for a sum. Therefore, it has all the properties of the sum

First, we have the distributive law:

\[c\sum_{i=1}^{n}a_i = \sum_{i=1}^{n}ca_i\]

where \(c\) is a constant. This is obvious once we expand the sums on both sides. That is,

\[c(a_1 + a_2 + \cdots + a_n) = ca_1 + ca_2 + \cdots + ca_n.\]

Next, we have the addition of two sums:

\[\sum_{i=1}^{n}a_i + \sum_{i=1}^{n}b_i = \sum_{i=1}^{n}(a_i + b_i).\tag{eq:sum2}\]

That is,

\[(a_1 + a_2 + \cdots + a_n) + (b_1 + b_2 + \cdots + b_n) = (a_1 + b_1) + (a_2 + b_2) + \cdots + (a_n + b_n).\]

This property stems from the commutativity of addition (i.e., we can change the order of terms in addition). Note that the upper and lower bounds of the index must match between the two sums. Also, note that we should use the parentheses in the body of the right-hand side. Otherwise, it looks like

\[\sum_{i=1}^{n}a_i + b_i\]

which may mean a completely different quantity:

\[\left(\sum_{i=1}^{n}a_i\right) + b_i.\]

This latter expression may or may not make sense depending on the context, but it definitely means something different from the expression in (eq:sum2) above.

If we multiply two sums, we obtain a double sum:

\[\sum_{i=1}^{n}a_i\sum_{i=1}^{m}b_i = \sum_{i=1}^{n}\sum_{j=1}^{m}a_ib_j.\]

Here, the left-hand side means the product of two sums:

\[\sum_{i=1}^{n}a_i\sum_{i=1}^{m}b_i = \left(\sum_{i=1}^{n}a_i\right)\left(\sum_{i=1}^{m}b_i\right)\]

Note that the index \(i\) in the first sum and that in the second sum on the left-hand side mean completely different things.

\[ \begin{eqnarray*} \left(\sum_{i=1}^{n}a_i\right)\left(\sum_{i=1}^{m}b_i\right) &=& \left(\sum_{i=1}^{n}a_i\right)\left(b_1 + b_2 + \cdots + b_m\right) ~~ \text{(expand the second sum)}\\ &=& \left(\sum_{i=1}^{n}a_i\right)b_1 + \left(\sum_{i=1}^{n}a_i\right)b_2 + \cdots + \left(\sum_{i=1}^{n}a_i\right)b_m ~~ \text{(distributive law)}\\ &=& \left(\sum_{i=1}^{n}a_ib_1\right) + \left(\sum_{i=1}^{n}a_ib_2\right) + \cdots + \left(\sum_{i=1}^{n}a_ib_m\right) ~~ \text{(distributive law)}\\ &=& \sum_{j=1}^{m}\sum_{i=1}^{n}a_ib_j\\ &=& \sum_{i=1}^{n}\sum_{j=1}^{m}a_ib_j ~~ \text{(change the order of sums)}. \end{eqnarray*} \]

When you are not sure, just try to expand it to see how it looks.


Common mistakes with the summation symbol.

Below are a few of the common mistakes I find among students. You should spot what is wrong with these.

\[\sum_{i=1}^{n}b = b. ~ \text{(Wrong!)}\]

\[\sum_{i=1}^{n}a_i\sum_{i=1}^{n}b_i = \sum_{i=1}^{n}a_ib_i. ~\text{(Wrong!)}\]

\[\sum_{k=1}^{n}ka_k = k\sum_{k=1}^{n}a_k. ~\text{(Wrong!)}\]



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