How to use the summation symbol

If we want to add the expression x_1,x_2 all the way up to x_{10}, it is quite cumbersome to write x_1 +x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 +x_9 +x_{10}. Mathematical notations permit us to shorten such addition using the \cdots symbol to denote “all the way up to” or “all the way down to”. Using the this symbol, the expression above can be written as x_1 + x_2 + \cdots + x_{10}.

There is, however, a more compact way of writing sums. We can use the Greek letter \Sigma as shown below.

Figure 1 – The Sigma Notation

In the figure above, is the first index, and letter b is the last index.   The variable(s) are the letters or the numbers that appear constantly in all terms. In the expression

x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 + x_{10}

1 is the first index, 10 is the last index and x is the variable. We use the letter i as our index variable, or the variable that will hold the changing quantities.  Hence, if we are going to use the sigma or the summation notation for the expression x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 + x_{10}, we have

\displaystyle\sum_{i=1}^{10} x_i

Some of the examples are shown below.  Observe the colors of the indices and the variables, to familiarize yourself how the summation symbol works.

Figure 2 – Examples of Summation Notation

In using the summation symbol, take note of the following:

  • An index variable is just a “dummy” variable. It means that you can use a different index variable without changing the value of the sum. The sum \displaystyle\sum_{i=1}^{10} a_i is the same as \displaystyle\sum_{j=1}^{10} a_j and is the same as \displaystyle\sum_{k=1}^{10} a_k.
  • The indices are the natural numbers 1, 2, 3, \cdots and so on.
  • The last index is always greater than the first index.
  • A variable without an index most of the time represent an infinite sum or a sum from 1 through n

More Examples

1. (a - 1) + (a^2 - 2) + (a^3 - 3) + (a^4 -4) \displaystyle\sum_{i=1}^{4} (a^i - i)
2. 3p_5 + 3p_6 + 3p_7 + 3p_8 \displaystyle\sum_{j=5}^{8} 3p_j
3. 5 + 5 + 5 + 5 + 5 + 5 + 5 \displaystyle\sum_{k=1}^{7} 5
4. 1 + 2 + 3 + \cdots + 99 + 100 \displaystyle\sum_{m=1}^{100} m
5. (a_3 + b_3) + (a_4 + b_4) +(a_5 + b_5) \displaystyle\sum_{n=3}^{5} (a_n + b_n)

Properties of the Summation Symbol

1.) The expression (x_1 + x_2 + x_3 + x_4) + (x_5 + x_6 + x_7 + x_8 + x_9 + x_{10}) equals (x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 + x_{10}) which means that \displaystyle\sum_{i=1}^{4} x_i + \displaystyle\sum_{j=5}^{10} x_j = \displaystyle\sum_{i=1}^{10} x_i . In general, \displaystyle\sum_{i=1}^{m} x_i + \displaystyle\sum_{j=m+1}^{n} x_j = \displaystyle\sum_{i=1}^{n} x_i.

2.)The expression (x_1 + x_2 + x_3 + x_4) + (y_1 + y_2 + y_3 + y_4) = \displaystyle\sum_{i=1}^{4} x_i + \displaystyle\sum_{j=1}^{4} y_j. Regrouping the expression, we have  (x_1 + y_1) + (x_2 + y_2) + (x_3 + y_3) + (x_4 + y_4) = \displaystyle\sum_{i=1}^{4} (x_i + y_i). This means that \displaystyle\sum_{i=1}^{4} x_i + \displaystyle\sum_{i=j}^{4} y_j = \displaystyle\sum_{i=1}^{4} (x_i + y_i) Generalizing, we have \displaystyle\sum_{i=1}^{n} x_i \pm \displaystyle\sum_{j=1}^{n} y_j = \displaystyle\sum_{i=1}^{n} (x_i \pm y_i).

3.) The expression c + c + c + \cdots + c (k of them) = \displaystyle\sum_{i=1}^{k} c. But c + c + c + \cdots + c = c( 1 + 1 + 1 + \cdots + 1) (k of them) = kc. Therefore, \displaystyle\sum_{i=1}^{k} c = kc.

4.) The expression 2x_1 + 2x_2 + 2x_3 + 2x_4 = \displaystyle\sum_{i=1}^{4} 2x_i. But 2x_1 + 2x_2 + 2x_3 + 2x_4 = 2(x_1 + x_2 + x_3 + x_4) = 2 \displaystyle\sum_{i=1}^{4} x_i.  In general, \displaystyle\sum_{i=1}^{k} cx_i = c \displaystyle\sum_{i=1}^{k} x_i.

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