Sigma Notation/ Convergent Series and sum to Infinity
Sigma Notation
When we look at adding a sequence there is a different and more mathematical way of
writing this.
Consider the following:
u1, u2, u3, … ,un is a sequence and we know that u1 + u2 + u3 + … +un is the series for this
sequence.
The mathematical way of representing this and what we will see in higher-level math
courses uses the Greek letter sigma (Σ) to represent the sum of values.
𝑛
∑ 𝑢𝑖
𝑖=1
This notation means to sum the first “n” terms of a sequence. It would read “the sum of
all the terms ui from i=1 to i=n”.
Consider the following arithmetic sequence 2, 6, 10, 14,… It has a starting value of 2 and
a common difference of 4. We can write this in the general form of un = 4n – 2.
In sigma notation we could write this as:
This would read “the sum of all the terms of _____________ from _______ to
_______”.
Calculating this would look like this:
4
Ex.
Write the expression
x
x 1
2
2
as a sum of terms, and then calculate that sum.
6
Ex.
Evaluate the expression
2
a
a 3
Ex.
Write the series 2 + 10 + 50 + 250 + 1250 + 6250 using sigma notation.
Convergent Series and sum to infinity
Let’s investigate the following geometric series:
a.
2 + 1 + 0.5 + 0.25 + … b. 75 + 30 + 12 + …
What is the common ratio, r?
a.
b.
c. 240 – 60 + 15 – 3.75 + …
c.
Find the following and write the full value of what’s on the GDC screen:
a.
b.
c.
S10 
S10 
S10 
S15 
S15 
S15 
S20 
S20 
S20 
Do we notice any patterns? What do we think is happening?
For each case, let’s find the value of S50 . Do we think this is as accurate value? Why or
why not?
These are all examples of convergent series. These types of series occur when
1  r  1 . The differences between each term decreases as n increases. This means that
as you add more terms, the value of the sum changes very little. The sum is actually
approaching some constant value as n gets very large.
The sum of the terms of a geometric series is Sn 
u1 1  r n 
1 r
,
As n gets very large, we say that n ‘approaches infinity’, or n   .
If 1  r  1 , then as n   , r n  0 , so S n 
u1 (1  0)
u
 1 .
1 r
1 r
 u1 1  r n   u
  1 , or for our case we will say
In future courses, we would write: lim 
1 r  1 r
n  


u1
that S  
1 r
This means that as n gets very large (it approaches infinity), the value of the series is
u
u
approaching 1 . The series is converging to the value 1 . We write this as
1 r
1 r
u
S   1 and call it ‘the sum to infinity’.
1 r
Ex.
For the series 18 + 6 + 2 + …, Find S10 , S15 and S .
Ex.
The sum of the first three terms of a geometric series is 148, and the sum to
infinity is 256. Find the first term and the common ratio of the series. (Do this on
the back of this sheet)
Worksheet