Summation
Sum thing you should know about.
Summation is the iterative addition of a sequence of numbers, denoted by \(\sum.\)
The summation of an expression \(a_i\) from \(i=m\) to \(n\) is written as:
$$\sum\limits_{i=m}^{n}a_i$$
Initially, \(i=m.\) Each iteration, \(i\) gets incremented by \(1\) until \(i=n.\) To evaluate a
summation, evaluate \(a_i\) for each value \(i\) can attain, and then sum it all up:
$$\sum\limits_{i=m}^{n}a_i = a_m + a_{m+1} + ... + a_{n-1} + a_n$$
Alternatively, the summation of an expression \(a_i\) for each element in a set \(S = \set{x_1, x_2, ...
x_k}\) would look like this:
$$\sum\limits_{i \in S}a_i$$
To evaluate this summation, evaluate \(a_i\) for each value \(i\) can attain, which in this case is each
element of \(S\), and then sum it all up as before:
$$\sum\limits_{i \in S}a_i = a_{x_1} + a_{x_2} + ... + a_{x_k}$$
Contents
Nested summations
When you see a summation within another summation, don't panic. It is important that you take things one step at a time, so that you don't mess up. The recommended strategy is to evaluate the innermost summation first, and work your way outward. For example:
Evaluate:
$$\sum\limits_{i=5}^{7}(\sum\limits_{j=2}^{3}4i - j)$$
My first step will be to expand the inner summation:
$$\sum\limits_{i=5}^{7}((4i - (2)) + (4i - (3)))$$
Notice that the variable \(i\) is not being evaluated yet because it is not the subject of the innermost
summation, \(j\) is. Next, simplify:
$$\sum\limits_{i=5}^{7}(8i - 5)$$
Now, let's deal with \(i\) and expand the outer summation:
$$(8(5) - 5) + (8(6) - 5) + (8(7) - 5)$$
Upon simplification:
$$129$$
And that is our answer. All done!