Science:Infinite Series Module/Units/Unit 1/1.3 Infinite Series/1.3.03 The Geometric Series

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There are certain forms of infinite series that are frequently encountered in mathematics. The following example

for constants and is known as the geometric series. The convergence of this series is determined by the constant , which is the common ratio.

Theorem: Convergence of the Geometric Series
Let and be real numbers. Then the geometric series 

converges if , and otherwise diverges.

Proof

To prove the above theorem and hence develop an understanding the convergence of this infinite series, we will find an expression for the partial sum, , and determine if the limit as tends to infinity exists. We will further break down our analysis into two cases.

Case 1:

If , then the partial sum becomes

So as

we have that

.

Hence, the geometric series diverges if r = 1.

Case 2:

A short derivation for a compact expression for will be useful. First note that

Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} s_n &= a + ar+ar^2+\ldots+ar^{n-1} \\  rs_n &= ar + ar^2+ar^3+\ldots + ar^{n-1} + ar^n. \end{align}}

The second equation is the first equation multiplied by . Subtracting these two equations yields

Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} s_n - rs_n &= a - ar^{n} \\ s_n &= \frac{a(1 - r^{n})}{1-r}, \ r\ne 1. \end{align} }

Using this result, we see that:

  • if , then as n → ∞, sna/(1-r)
  • if , then as n → ∞, sn → ∞
  • if , then the series is divergent by the Divergence Test (which we cover in a lesson in Unit 2)

Summary

The above results can be summarized in the following figure.

thumb