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## Integral Calculus

### Course: Integral Calculus>Unit 5

Lesson 2: Infinite geometric series

# Worked example: convergent geometric series

AP.CALC:
LIM‑7 (EU)
,
LIM‑7.A (LO)
,
LIM‑7.A.3 (EK)
,
LIM‑7.A.4 (EK)
Sal evaluates the infinite geometric series 8+8/3+8/9+... Because the common ratio's absolute value is less than 1, the series converges to a finite number.

## Want to join the conversation?

• Where can I find the video that derives the formula for the sum of infinite geometric series?
• I see you were given a link. The proof is very simple:
𝑆 = 𝑎₁ + 𝑎₁𝑟 + 𝑎₁𝑟² + ...
𝑆 = 𝑎₁ + 𝑟(𝑎₁ + 𝑎₁𝑟 + ...)
𝑆 = 𝑎₁ + 𝑟𝑆
𝑆 – 𝑟𝑆 = 𝑎₁
𝑆(1 – 𝑟) = 𝑎₁
𝑆 = 𝑎₁/(1 – 𝑟)
• Can sigma notation and S sub n be used interchangeably?
• No, sigma notation is something TOTALLY different. S sub n is the symbol for a series, like the sum of a a geometric sequence. Sigma notation is just a symbol to represent summation notation. Honestly, I see what you mean, about them seeming the same, but I don't think it could be used interchangeably.
• Am I visualizing this correctly?

So let's suppose we have a piece of bread that is 12 cm long. For the first slice, I cut it at exactly 8 cm and the slices that follow are one-third the length of the previous slice, does that mean that I will never run out of bread to divide given that I have the ability to precisely slice it at the correct length?
• In the situation you describe, the lengths can be represented by the 8 times the geometric series with a common ratio of 1/3. The geometric series will converge to 1/(1-(1/3)) = 1/(2/3) = 3/2. You will end up cutting a total length of 8*3/2 = 12 cm of bread. So, you will never run out of bread if your first slice is 8cm and each subsequent slice is 1/3 as thick as the previous slice.
(1 vote)
• Would anyone happen to know if there any videos on arithmetic sequences?
• Why is the limit of (1/3)^k at k = infinity not 0?
(1 vote)
• The limit is 0. It is the sum of the series that coverges to another value as k approaches infinity.