Infinite Geometric Series

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

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.

Create a free Studyio account to take the full lesson with voice-over teaching, practice questions and instant feedback.