A sequence of numbers is said to be geometric if any term after the first can be obtained by multiplying the previous number by the same constant. This constant is called the common ratio. So the sequence 1,2,4,8,16 is geometric since each number in the sequence after the first can be obtained by multiplying the previous term by 2. Here the common ratio is 2. A geometric series is the sum of the terms of a geometric sequence. Thus, 1+2+4+8+16 is a geometric series. Geometric series may be finite, such as the series in the preceding sentence, or infinite, such as 1+2+4+..., where the three dots indicate that the series follows this pattern forever. A finite series obviously has a sum, such as 1+2+4+8+16=31. If we write the general finite geometric series as a+ar+ar2+ar3+...+arn=Sn, then it can be shown that Sn=a-arn+1)/(1-r). Trying this formula for 1+2+4+8+16, we get (1-25)/(1-2)=(-31)/(-1)=31, agreeing with our answer computed in the traditional manner above. What may be surprising is that some infinite geometric series also have finite sums. This is true whenever the absolute value of r is less than 1, or equivalently, when -1 < r < 1.
Here's why. When -1 < r < 1, the expression rn+1 approaches 0 when n is very large, so for an infinite geometric series the expression for the sum a-arn+1)/(1-r) approaches a/(1-r).
The ancient Greek philosopher Zeno (5th century BC) was famous for creating paradoxes to vex the intellectuals of his time. In one of those paradoxes, he says that if you are 1 meter away from a wall, you can never reach the wall by walking toward it. This is because first you have to traverse half the distance, or ½ meter, then half the remaining distance, or ¼ meter, then half again, or 1/8 meter, and so on. You can never reach the wall because there is always some small finite distance left. The theory of infinite geometric series can be used to answer this paradox. Zeno is actually saying that we cannot get to the wall because the total distance we must travel is ½ + ¼ + 1/8 + 1/16 +..., an infinite sum. But according to our discussion in the preceding paragraph, this is just an infinite geometric series with first term ½ and common ratio ½, and its sum is (1/2)/(1-1/2)=1. So the infinite sum is one meter and we can indeed get to the wall.
Infinite geometric series are of major importance in calculus in connection with Taylor Series. A Taylor Series is an infinite series representation of some mathematical function. Many of the most important mathematical functions, such as sin x, cos x, ex, ln x and others may be expressed as infinite Taylor series, which is useful in creating algorithms for calculators and computers to give very accurate approximations of these functions for specific values of x. Referring again to the opening paragraph of this article, we can see that 1/(1-x)= 1+x+x2+x3+..., whenever -1 < x < 1, and this is just the Taylor series for the function 1/(1-x). This series can then be used to derive series for other important mathematical functions.
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