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Geometric Series and the Ratio Test

Geometric Series and the Ratio Test

Geometric Series and the Ratio Test

Geometric Series and the Ratio Test

Geometric Series and the Ratio Test

Geometric Series and the Ratio Test

A geometric series is a series of the form ar n (where we take a and r to be positive). One learns in high school algebra that this series converges if and only if 0 < r < 1 . If ar n does converge, we have

ar n =    

We can combine these comments about geometric series with the comparison test to yield another test called the ratio test: given a series a n with a n > 0 for all n , if there exists a number C with 0 < C < 1 such that

C    

for all n , then a n converges.

To prove this fact, note that under the hypotheses of the theorem,

a nCa n-1C 2 a n-2 ... C n-1 a 1    

Letting b n = a 1 C n-1 , so that b n is a (convergent) geometric series, we see that a nb n . By the comparison test, a n must also converge; in fact

a n b n =