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The animation to the right shows a sequence of purple rectangles appearing one by one, slowly filling up the inside of a larger square around the border. The first rectangle has 21 the area of the border square, the second has 41 the area, the third 81, and so on.
Can you describe the total area covered by purple rectangles when there are
two rectangles?
three rectangles?
four rectangles?
Now consider what happens when we continue adding purple rectangles for a long time. We can tell that the white, "uncovered" area gets smaller and smaller with each additional purple rectangle that appears. However, it's getting cut in half each time, not shrunk by a constant amount. So will there ever come a point where there's no white area at all, and the rectangles cover up the entire unit square?
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This question is getting at the central idea of geometric series. Before we try to answer it in the most extreme case -- infinite rectangles -- let's work through some simpler examples.
Checkpoint
Consider the sequence
which is geometric with u1=1 and r=2.
What is the sum of the first four terms in the sequence 1,2,4,8…?
Select the correct option
It's simple enough to find the sum of the first four terms of a sequence: just add them all up by hand. But what if we wanted to find the sum of ten terms? Of twenty? Of a hundred?
Clearly, we're going to need a better way to do this. Let's work through an abstract version of this type of sum and see if we can come up with a more general formula.
Discussion
Let Sn denote the sum of the first n terms of a geometric sequence. The sum of the first four terms of an arbitrary geometric sequence, with first term u1and common ratio r, is thus written
First, write an expression for rS4 (the sum of the first four terms multiplied by r).
Now write an expression for rS4−S4. Simplify it as much as you can.
Finally, use your answer to part (b) to write an alternative expression for S4. (Hint: rS4−S4=(r−1)S4)
Part (a)
We have
so multiplying both sides by r gives
Part (b)
We have from part (a):
Hence
Part (c)
From part (b) we have
so
Finite Geometric Series
The sum of the first n terms in a geometric sequence is given by:
Exercise
A geometric sequence has first term u1=81 and fourth term u4=3.
Find the sum of the first five terms.
Select the correct option
We've seen that geometric series can evaluate to very large values very quickly, even if their common ratio is quite small, say r=2. This is because of the rapidly growing behavior of exponential functions: 2 might be small, but 210 sure isn't!
Something interesting happens, though, when we consider geometric series with fractional common ratios like 21 or −31, since raising these values to consecutive exponents makes them shrink instead of grow. Let's play with small common ratios some more to learn more about the behavior of these types of geometric series.
Discussion
For a geometric sequence with first term u1=1 and common ratio r=1.1, find S10 and S50.
For a geometric sequence with first term u1=1 and common ratio r=0.9, find S10 and S50.
State the main differences between the results of (a) and (b). Specifically, how do the values of S10 compare to the values of S50 in each case?
Part (a)
For a geometric sequence with first term u1=1 and ratio r=1.1, the sum of the first n terms is
Hence
on a TI-84 you would enter
Similarly,
enter
Part (b)
For a geometric sequence with u1=1 and r=0.9, the sum of the first n terms is
Hence
Calculating (to four decimal places):
Part (c)
For r=1.1, the sum of the first 10 terms is approximately 15.94, while the sum of the first 50 terms is approximately 1163.91:
This means that the sum after 50 terms is much larger than after 10 terms—about 75 times greater. Even though each term is only 10% larger than the previous one, this repeated multiplication causes the sum to grow extremely quickly as more terms are added. The total does not level off; it keeps increasing without bound.
For r=0.9, the sum of the first 10 terms is approximately 6.51, and the sum of the first 50 terms is approximately 9.95:
Here, the sum after 50 terms is only a little larger than after 10 terms. Most of the total sum is reached very quickly, and adding more terms makes only a small difference. This is because each term is only 90% of the previous one, so the terms get smaller and smaller, and the sum approaches a limiting value.
It turns out that while geometric series with ∣r∣>1 (like r=1.01,−5, or 7.5) keep growing and growing infinitely, geometric series with ∣r∣<1 (like r=−0.99,0.5, or 0.113) approach a finite value. What value they approach also follows a rule:
Infinite Geometric Series
If a geometric sequence has a common ratio ∣r∣<1, then each term will be smaller than the previous term. As the terms get smaller and smaller, the sum of all the terms approaches a finite value:
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Checkpoint
A geometric series has first term u1=1 and common ratio r=0.9.
Find S∞.
Select the correct option
It's crucial to remember that no matter how small r is, if its absolute value is greater than 1, the series that takes its sum will grow infinitely. Even a geometric series with r=1.000000000001 has S∞→∞!!
Convergence
A geometric series is said to converge if S∞ is finite - which means ∣r∣<1⇔−1<r<1.
Example
A geometric sequence has u1=8 and u4=2k+1. For what value(s) of k does the corresponding geometric series converge?
We have
Now if −1<r<1, then −1<r3<1:
Exercise
A geometric sequence has consecutive terms 3 and 2k−1. Find the possible value(s) of k such that S∞ exists.
Select the correct option
Discussion
Remember those rectangles with shrinking area from the beginning of the lesson?
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Will they ever completely cover the area inside the large border square?
In the context of this "real-world" scenario, does it make sense to talk about what happens as we approach infinitely many rectangles?
At each step the new rectangle has half the area of the previous one, so the covered area after n rectangles is a partial sum of a geometric series.
Since this is a geometric series with first term a=21 and ratio r=21,
Because 2n1>0 for every finite n, we have Sn<1. In other words no finite collection of rectangles ever covers the whole square. However,
so in the limit the infinitely many rectangles “fill” the square.
In a real‐world setting, though, we cannot place infinitely many ever‐smaller rectangles (matter cannot be subdivided below atomic scales, we cannot draw infinitely many shapes in finite time, nor detect arbitrarily tiny regions). Thus the infinite process is a useful mathematical idealisation but has no literal physical counterpart.
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