An Introduction to Summation Notation
(or, Everything About Summation Notation
You Need for the Test, and Then Some)
Remember that summation notation (using ∑ to write a sum rather than a
bunch of + symbols) is basically a convenient form of shorthand. For instance, the
following sum
is read as “the sum over i from one to five of two i”, and is a shorthand for the
sum 2(1) + 2(2) + 2(3) + 2(4) + 2(5), which can also be written as 2 + 4 + 6 + 8 + 10.
Therefore we may write
As another example, we have
7
i
2
 12  2 2  3 2  4 2  5 2  6 2  7 2  140
i 1
We also might want to go the other way, that is, given a sum like 2+5+8+11+14,
we might want to write the sum in sigma notation. In this case we have five
terms, so we would start off with, say,
We now want to find the function f that belongs to the right of the sum. Looking
at our series, we see that the first term must be 2, so we require that f(1) = 2.
Similarly the second term must be 5, so we require that f(2) = 5. We also require
that f(3) = 8, f(4) = 11, and f(5) = 14. What sort of function meets these
requirements?
Maybe the answer is clear, but if not we might try graphing the ordered
pairs (i, f(i)) and see if we detect a pattern:
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Y-Values
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Clearly this is a linear function, and if it’s still not clear from the graph what
function this is, a bit of analysis will show us that the slope is 3 (because, e.g.,
) and that the y-intercept is -1. Therefore this function is
, and we can now conclude that
Of course, we could have written this sum in a number of different ways;
for instance, if we were feeling particularly masochistic, we could have written
although since the coefficients were rounded to three decimal places the answer
won’t be exact (but it could be made so, if we really wanted to put in the effort).
In fact, we could in theory find an infinite number of functions for which f(1) = 2,
f(2) = 5, …, f(5) = 14, and any of them would be a correct answer.
Nor was it necessary that we start with i=1. We could have selected any
value of i. For instance, if we started from i=0 we would have that f(0) = 2, f(1) =
5, …, f(4) = 14. We could plot those points as well:
Note that the slope is still the same, but now the y-intercept is 2. Thus the
desired equation is now f(x) = 3x+2, and we have
Neither of the forms
and
is better than the other;
both are completely accurate, and by changing the indices to other numbers we
could alter the look yet again, but all of them would still describe the same sum.
Now that we’re more comfortable with summation notation, we might
want to answer a few questions:
Q: What if i doesn’t appear in the sum, e.g., as in
?
A: No problem. In this case
[the constant function] and we have f(1)=3,
f(2)=3, f(3)=3, f(4)=3, and so on. Thus we have
Q: Do we have to use i as the index?
A: No. In fact, we frequently use j or k as an index as well. For instance, if we
were working with complex numbers, where i could refer to the square root of
negative one, it would be confusing and improper to have i as an index as well.
Mathematically
and
are exactly equivalent.
Q: What about when we have a constant like n in our summation, e.g., as in
?
A: We treat it just like we would any other constant like, for instance, 10. The
only difference is that we can’t write out the entire expression without using
either sigma notation or ellipses. We might write the above as
And that’s the best we can do, unless we do a bit of algebraic manipulation to
show that
. The form
is called a closed-
form expression, and is usually preferable to either half of the equation above,
but unfortunately closed form expressions (a) don’t always exist and (b) aren’t
always easy to find. For instance, most of the time we will be using summation
notation in this class to refer to random data points which, by virtue of their
randomness, would not be expected to follow some nice, easily described
function.