Convergence/Divergence Series
A Traveler's Paradox
Let's say a traveler is driving a car down a road that is one mile long. At the start, the car is moving 60 miles per hour. After one second, the car immediately reduces its speed to 30 miles per hour. After two seconds, the car instantaneously reduces its speed from 30 miles per hour to 15 miles per hour. More generally if v(t) is the speed of the car at time t, then
This is where n = 1, 2, 3, 4, 5,......
In the formula, the time t is in seconds, and the speed v(t) is in feet per second, and vn is the distance traveled after n seconds.
The distance traveled after n seconds takes the form of a geometric sum. This sum simplifies to
By passing this equation, which we've called (1), to the limit as n tends to infinity, we can then see that the car's distance will approach 176 feet as the time t tends to infinity. However since the expression given in (1) is strictly less than 176, the car will never reach the end of the road. Instead, the car slows down so drastically that it gravitates towards the 176 foot marker. The marker is seen as a point of attraction along the one mile road. It appears counter-intuitive that even though the car never reduces its speed to zero, it never comes anywhere close to reaching the end of the road - it's not even one-thirtieth of the distance!
Figure 1 illustrates the car's traveling progression. The first car represents the position at zero seconds, the second car at one second, the third at two seconds, and so on (from left to right). As is apparent, the one second increments decrease with time. We can also see this from the identity
The moral of this story is that a sum having an infinite number of terms can possibly end up being finite. We've given an example of a convergent geometric series, making the concept of a convergent series more precise.
The Theorem
To begin, we define the theorem, which we'll call equation 2:
where r does not equal 0 if the limit of a partial sum exists as a real number. We write the definition of an infinite series, like this one, and say the series, like the one here in equation 3, converges.
If the aforementioned limit fails to exist, the very same series diverges. It's denoted as an infinite sum whether convergent or divergent.
The partial sums in equation 2 are geometric sums, and this is because the underlying terms in the sums form a geometric sequence. The respective infinite series is then called a geometric series.
Which values of r does the series given in (3) converge? It turns out that there is a convenient, shortcut formula for the partial sums defined in (2). The formula can be proven directly by induction, and for that reason, we omit the proof here.
Here's the Lemma: Geometric Sum Formula in equation 4:
For n = 0, 1, 2, 3, 4, ....
As an immediate consequence of equation 4, we arrive at the geometric series theorem. This theorem gives the value of r for which the series converges and diverges.
We first let the partial sums be defined as in (2) and the respective infinite series defined as in (3). Then, we have the following results:
In this case, we get our geometric series formula and case ii:
In this second case, we get the following resulting sum value in case ii and the divergent oscillation in case iii.
In this third and final case, the series cannot be assigned any value, since the limit of the partial sums does not exist even in the extended real number sense. The monotone convergence theorem ensures the existence of the limits in cases i and ii (as extended real numbers).
A Traveler's Paradox: Revisited
We now generalize the preceding example to the following. Suppose for n = 1, 2, 3, 4, 5, .... we use the general form of the velocity function where r does not equal 0:
Then by (4), the position after n seconds is
By using the fact that the velocity is constant between successive integer seconds, the position at time t is seen to be defined by the formula appearing here:
Note that negative r allows the car to move in reverse. Then depending on whether the absolute value of r is less than one, the vehicle's position will either hone in on a fixed location or exhibit increasingly larger distances from the initial location. You can see how this is all played out with the tables and figures that appear. You don't have to memorize all of the content in these tables and figures; it's just important that you see all of the data in one place:
Here's Table 1a in which r = 0.5:
| Elapsed Time (Seconds) |
Distance Traveled (Feet) |
Distance Increment (lag one difference) |
0
1
2
3
4
5
6
7
8
9
10 |
0
88
132
154
165
170.5
173.25
174.625
175.3125
175.65625
175.828125 |
88
44
22
11
5.5
2.75
1.375
0.6875
0.34375
0.171875 |
Here's Figure 2a, in which r = 0.5:
Now, here's Table 1b, in which r = -0.5:
| Elapsed Time (Seconds) |
Distance Traveled (Feet) |
Distance Increment (lag one difference) |
0
1
2
3
4
5
6
7
8
9
10 |
0
88
44
66
55
60.5
57.75
59.125
58.4375
58.78125
58.609375 |
88
-44
22
-11
5.5
-2.75
1.375
-0.6875
0.34375
-0.171875 |
Now, here's Figure 2b, in which r = -0.5:
Now, here's Table 1c, in which r = -2:
| Elapsed Time (Seconds) |
Distance Traveled (Feet) |
Distance Increment (lag one difference) |
0
1
2
3
4
5
6
7
8
9
10 |
0
88
-88
264
-440
968
-1848
3784
-7480
15048
-30008 |
88
-176
352
-704
1408
-2816
5632
-11264
22528
-45056 |
Now, here's Figure 2c, in which r = -2:
Now you should understand and be able to actually see how both convergence and divergence of a series can play out.
Lesson Summary
Let's take a couple of moments to review what we've learned. We learned that a geometric series is simply a constant ratio between terms that follow one another. We gained our insight of a geometric series by studying a vehicle's movement along a one-dimensional axis. The partial sums form a sequence of real numbers which tends to a real number when convergent. If this does not occur, the geometric series diverges. The series can diverge in two different ways, and this depends on whether r is positive or negative.
We also learned that the geometric series theorem gives the value of r for which the series converges and diverges. The concept of convergence/divergence extends to a broader class of series. The geometric series is one of the few series where we have a formula when convergent that we will see in later sections.
