Infinite Geometric Series

Find sums of infinite geometric series using convergence. Math 20-1 Alberta curriculum.

Lesson 3.5 of Sequences and Series in Math 20-1: Alberta curriculum lessons.

What you'll learn in this lesson

When Does an Infinite Sum Exist?. Discover what makes an infinite geometric series converge or diverge by comparing two series side-by-side, then learn the convergence condition based on the common ratio.
The Infinite Sum Formula. Derive and apply the infinite sum formula, confirm it with a visual intuition diagram, and use it to convert a repeating decimal into an exact fraction.
Working Backwards and Applications. Apply the infinite sum formula to real-world problems, find missing first terms or ratios algebraically, and determine the values of a variable that make a series converge.

Infinite Sequence and Series. An infinite sequence is a sequence with an unlimited number of terms: t_1, t_2, t_3,

An infinite geometric series is a geometric series that goes on forever: t_1 + t_2 + t_3 +

For example, the series 4 + 2 + 1 + 0.5 + 0.25 + never stops. The + means we keep adding terms indefinitely.

Let's see what happens to the partial sums of two different infinite geometric series.

Series 1: 4 + 2 + 1 + 0.5 + 0.25 + where t_1 = 4, r = (1)/(2). S_4 = 4 + 2 + 1 + 0.5 = 7.5

S_6 = 4 + 2 + 1 + 0.5 + 0.25 + 0.125 = 7.875

S_7 = 7.9375

S_75 ≈ 8 (using the formula)

S_100 ≈ 8 (using the formula)

The partial sums get closer and closer to 8. Even at S_75 and S_100, the result rounds to 8. The series settles on a fixed value.

Series 2: 4 + 8 + 16 + 32 + 64 + where t_1 = 4, r = 2. S_4 = 4 + 8 + 16 + 32 = 60

S_6 = 252

S_7 = 508

S_20 = 4 194 300

S_30 = 4 294 967 292

The partial sums grow without bound. They keep getting bigger forever. The series does NOT settle on any fixed value.

Why the Difference?. The difference comes down to r, the common ratio.

In Series 1, r = (1)/(2). Each term is smaller than the one before. The added pieces shrink toward zero, so the total approaches a limit.

In Series 2, r = 2. Each term is bigger than the one before. The added pieces grow without bound, so the total explodes.

Part a) 0.5 - 1 + 2 - 4 + . Step 1: Find r:

r = (-1)/(0.5) = -2

Step 2: Check the condition:

|r| = |-2| = 2, which is greater than 1.

This series is divergent, no infinite sum exists.

Part b) 12 + 3 + (3)/(4) + . Step 1: Find r:

r = (3)/(12) = (1)/(4)

Step 2: Check the condition:

|r| = (1)/(4), which is less than 1.

This series is convergent, an infinite sum exists.

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