Definition of a Series
Let an be a sequence then we define the nth
sum of an as
an = n2 - 5
We just plug in 1 through 5 for n and add
S4 = (12 - 5) + (22 - 5) + (32 - 5) + (42 - 5)
= -4 + (-1) + 4 + 11 = 10
We can also write the problem in sigma notation as
We define the infinite series S as the limit of the sn that is
We write out the first four terms:
= -1/1 + 1/4 - 1/4 + 1/9 - 1/9 + 1/16 - 1/16 + 1/25 - ...
Such a series is called a telescoping series.
We define a geometric series to be a series of the form
The series below is an example of a geometric series.
S = 3/2 + 3/4 + 3/8 + ...
For geometric series we have the following theorem.
The formula comes directly from the formula for the sum of a finite geometric series.
If |r| < 1, then rn+1
goes to zero as n goes to infinity, and the formula follows.
The Limit Test
Caution: If the limit goes to zero then the series still may diverge.
Recently, the federal government gave a tax refund of a total of 10 billion dollars. Economists predict that Americans will spend 70% of this refund on items that directly go back into the wallets of Americans. Again 70% of the spent money will again be used toward money that can be spent in America. If this process continues forever at the 70% rate, how much money will exchanged from this original 10 billion dollars?
This is an infinite geometric series with
a = 10 and r = 0.7
We have that the sum is
There refund will result in 30 billion dollars being exchanged in the US.
The Harmonic Series
The last example is important enough to deserve its own theorem. We call
it the "Harmonic Series Test".
Proof: we write
1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 +
< 1/1 + 1/2 + 1/4 + 1/4 + 1/8 + /18 + 1/8 + 1/8
= 1/2 + 1/2 + 1/2 + ...
which diverges. Hence the harmonic series diverges.