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Infinite Geometric Series

Imagine the infinite ways you can top a hotdog. Relish looks kind of bland after you consider what strawberry sauce and sprinkles could do to a hotdog.

Infinite geometric series, which are just specially dressed finite geometric series, have some wacky properties that can make them interesting. The standard infinite geometric series looks like

We already know for a geometric series the nth partial sum is

where r is the ratio of consecutive terms, a is the first term, and n is the number of terms.

Sample Problem

Use the formula for Sn to find


(a) |r| < 1

(b) 1 < |r|

(c) r = 1 (hint: look at the expanded geometric series)

(d) r = -1 (hint: look at the expanded geometric series)


(a) Using the formula for Sn,

As n approaches ∞, since |r| < 1 the quantity rn approaches 0.

(b) Again we use the formula for Sn. Since |r| > 1, the quantity rn approaches ∞ as n approaches ∞.

The numerator of the fraction will get farther from 0 without bound as n approaches ∞, so this limit doesn't exist.

(c) Following the hint, look at the expanded series. When r = 1 the geometric series looks like

Since every power of 1 is 1, this is the constant series

a + a + a + ...

The nth partial sum is Sn = a · n.

The limit

does not exist. We found this result earlier when looking at a constant series

(d) Looking at the expanded series, when r = -1 we get

which is the alternating series

aa + aa + aa.

The partial sums are

S1 = a = a

S2 = aa = 0

S3 = aa + a = a

S4 = aa + aa = 0

and so on.

Since the partial sums bounce back and forth between two values,

does not exist.

The last example covered all cases for r. We weren't playing the role of Oedipus here, leading you blindly down a dangerous path. We know that the sum of an infinite series, if it exists, is the limit of the partial sums Sn. In the previous exercises we found that

only exists when |r| < 1, in which case

This means when |r| < 1, the sum of the infinite geometric series


In all other cases (when |r| ≥ 1) the limit of partial sums

doesn't exist. This means when |r| ≥ 1 the sum of the infinite geometric series

doesn't exist either.

All of this means that an infinite geometric series converges when the ratio r has magnitude strictly less than 1. An infinite geometric series diverges in all other cases.

Finding the sum of an infinite geometric series is easier than finding a partial sum, because we only need to know a and r. We don't need to worry about how many terms there are. There are infinitely many.

The sum of a convergent geometric series is

where a is the first term of the series and r is the ratio.

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