The Geometric Series

The Geometric Series

 

The geometric series is a sum of terms where each progressive term geometric series is multiplied by the same common ratio.

S = 1 + x + xx +…

We can solve for the sum of the series by doing some algebra with S

Sx = x + xx +…

Now if we subtract the two equations we find that

S-Sx = 1

So a simple formula for S is just

S = 1/(1-x)

Geometric series are an important example of infinite series. This means if you add up an infinite list of numbers but you get out a finite value which is called convergence. It is important to note that our formula for the geometric series is not always true. Imagine that the number x were 2 then the sum would turn out to be infinity because

2+4+8+16 +… is infinity

It turns out that the allowed values of x are between negative one and positive one. This is called the radius of convergence, the interval for which the sum works out.

As a final example we can describe how the geometric series can be applied to the problem of a bouncing ball. A ball starts its fall from a height h and then bounces back up to a heart rh where r is the coefficient of restitution. This is a measure of the inelasticity of the collision of the ball with the ground. Now when the ball bounces again will rise back up to a height rrh and so on. The total distance travelled by a ball is then

L = 2h +2hr + 2hrr+2hrrr+… = 2h/(1-r)

If you ever wondered what happens when a ball stops bouncing an it is about to quit its motion this answers your question to how far it travels. How about how long does it take to stop bouncing?

We know from physics that if you throw a ball up to a height h from the ground then the time to fall back down is just two times the square root of (2h/g). We can call this time T. So the total time for the ball to stop bouncing is

 

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