A series is
a set of numbers such as:
1+2+3
which has a sum. A series is sometimes called a progression, as in
"Arithmetic Progression".
A sequence, on the other hand, is a set of numbers such as:
2,1,3
where the order of the numbers is important. A different sequence from
the above is:
1, 2, 3
A series such as:
1+2+3...
has the same sum as:
2+1+3
but the numbers are in a different sequence.
Arithmetic
Series
Young
Gauss and The Sum of the Natural Numbers
Gauss
told the story that when he was a boy, the teacher ran out of stuff to
teach and asked them, in the remaining time before playtime, to compute
the sum of all the numbers from 1 to 20 (or similar... actually, the numbers were 1 to 40!).
Gauss
thought that 1+20 is 21. And 2+19 is also 21. And this is true for all
the similar pairs, of which there are 10. So... the answer is 210.
One
can wonder what would have happened had the teacher asked for the sum
of the numbers from 1 to 19. Perhaps Gauss would have noted that 1+19
is 20, as is 2+18. This is true for all the pairs, of which there are
9, and the number 10 is left on its own. Nine 20's is 180 and the
remaining 10 makes 190.
Or perhaps he would have thought the sum to 20 adds up to 210, and 20
less is 190.
The
Sum of the Natural Numbers, using the Gauss Trick
Let us write the sum of the natural numbers up to n in two ways as:
S_{n}=1+2+3+...+(n-2)+(n-1)+n
Sn=n+(n-1)+(n-2)+...+3+2+1
If we add these two series we get:
2S_{n}=(n+1)+(n+1)+...+(n+1)
There are n of these (n+1)'s, so
2S_{n}=n(n+1)
So:
S_{n}=n(n+1)/2
The
sum of the natural numbers from 1 to n is therefore half the product of
the first term plus the last one multiplied by the number of terms.
General
Arithmetic Series
A pure
arithmetic series is one where the difference between successive terms
is a constant. We can call the constant d. If the first term is a, then
the arithmetic series is:
a+(a+d)+(a+2d)+...+(a+(n-1)d)
Using the Gauss trick, and writing this series in two different ways:
S_{n}=a+(a+d)+(a+2d)+...+(a+(n-2)d)+(a+(n-1)d)
S_{n}=(a+(n-1)d)+(a+(n-2)d)+...+(a+2d)+(a+d)+a
Adding the corresponding terms, noting they add up to 2a+(n-1)d:
2S_{n}=2a+(n-1)d+...+2a+(n-1)d
There are n of these terms, so:
2S_{n}=n(2a+(n-1)d)
S_{n}=n(2a+(n-1)d)/2
The
first term in the series is a, and the last one is a+(n-1)d, so we can
say the sum of the series is the first term plus the last term
multiplied by the number of terms divided by 2.
Geometric Series
A pure geometric series or geometric progression is one where the
ratio, r, between successive terms is a constant. Each term of a
geometric series, therefore, involves a higher power than the previous
term.
Algebraically, we can represent the n terms of the geometric series,
with the first term a, as:
S_{n}=a+ar+ar^{2}+ar^{3}+...ar^{n-1}
[1]^{
Each term is the previous term times r, so we can try multiplying the
series by r
}rS_{n}=ar+ar^{2}+ar^{3}+...+ar^{n-1}+ar^{n
}[2]
Subtracting Equation 2 from Equation 1, we get:
(1-r)S_{n}=a-ar^{n}
So, the sum of n terms of a geometric series with starting value a,
ratio, r is:^{
}
Probably because of the financial (compound interest) applications of
the geometric progression, the formula is written assuming that r is
less than one, but if r is greater than 1, then the minuses cancel out.
Bonus
If n is infinite and |r|<1, then r^{n}=0:
If a=1, we can note that:
So without dividing, and without using the Binomial Theorem, we get an
expression for (1-r)^{-1}
Simple
Arithmetic-Geometric Series
Consider the series:
This series is neither arithmetic (the differences between the terms
isn't constant) nor geometric (the ratio of successive terms isn't
constant), yet it seems to be something of both.
It looks like something that is familiar (1, 2, 3, ) yet alien.
If we know:
then we know the sum to infinity of the series is (1-r)^{-2},
if |r|<1 so the series converges. However, this doesn't tell us
the sum to n terms. Consider: [4.1] And using our trick from the geometric series, multiply this by r: [4.2] Subtract Equations 4.1, 4.2:
Noting we know the formula for the geometric series, and using it:
Bringing it all together under one denominator:
Rounding up like terms, gives us the formula:
Therefore:
If we multiply throughout by a constant, a, we get:
The sum to infinity of this series, when n tends to infinity (and |r|<1), is:
Simple Geometric Series 2
We
can write the series as in the following table. The top line (in bold)
is the series we are considering, and the lower lines are parts of that
series, put in the form of a normal geometric progression, so we know
how to sum them. All the lower series add up to the series we want to
sum.
Series Number
Terms
Sum
Number of Terms
1
+2r
+3r^{2}
+4r^{3}+
...
1)
1
+r
+r²
+r³
...
+r^{n-1}
n
2)
+r
+r^{2}
+r³
...
+r^{n-1}
n-1
3)
+r^{2}
+r³
...
+r^{n-1}
n-2
4)
+r³
...
+r^{n-1}
n-3
...
n)
+r^{n-1}
1
What we have done is split the series, which we do not know how to sum,
into a number of series which we do know how to sum. Each of these
series is one shorter than the previous.
And adding like terms, we get the formula for the series:
Therefore:
If we multiply throughout by a constant, a, we get:
The sum to infinity of this series, when n tends to infinity (and |r|<1), is:
This gives us another bonus, showing that (1-r)^{-2} gives our series (without using the Binomial Theorem, or polynomial division).
General Arithmetic-Geometric Series
In this series, which is neither geometric nor arithmetic, has the form:
The simple arithmetic-geometric series is a special case of this, where a=1. If we expand this series, we get: [5.1] Naturally,
we note the first bit is a normal geometric series, and the second bit
is our simple arithmetic-geometric series, which we have summed in the
previous section.
Now, as we have done all the work with the simple arithmetic geometric series, all that remains is to substitute our formula, (Noting that here, the number of terms is n-1) And to substitute the formula for the sum of a geometric series, into Equation 5.1 above:
That is:
Graph of Arithmetic, Geometric and Arithmetic-Geometric Progressions
History Note
Most of the stuff on this page was known over 2000
years ago by the Ancient Egyptians and Babylonians. The sums were
mentions in Euclid's Elements (about 2,300 years ago).