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### Course: Algebra 2 (Eureka Math/EngageNY)>Unit 3

Lesson 12: Topic E: Lesson 29: Geometric series

# Geometric series with sigma notation

A geometric series is the sum of the terms of a geometric sequence. Learn about geometric series and how they can be written in general terms and using sigma notation. Created by Sal Khan.

## Want to join the conversation?

• Is there a reason why "k" was used for the geometric series notation instead of "i"?
• You can pick any index variable you like. Common choices are i, k, and n.
• Why is the exponent on the r, k? Shouldn't it be (k-1) because the power to which the r is being raised is one less than the term, k, right?
• Value of k for the first term is defined under the sigma.
Since there is k = 0 under the sigma, the value of k in the first term will be 0. Value of k is increased by 1 for every next term.
• In sigma notation is it possible for the index to be a fraction or a decimal? If it is then how does it increase? Does it still increase by 1 ?
• No, the index variable always increases by 1. If you to increase by some other constant, just multiply the index variable that appears in the formula by the necessary number to produce the number you need to increase by.

Suppose you have (x²+7) and need to sum over x=0 to 10 in steps of ½. here is the notation for that:
∑ {(½x)² + 7}, over x=0 to 20
Notice the change in the bounds.
Also note it wouldn't be too hard to simplify this formula before doing the sum.
• Is it valid to take the common factor (here a ) out of the summation notation ?
• Yes. Note that Sum (ar^k) for k = 0 to n = a + ar + ar^2 + ... = a(1 + r + r^2 + ...) so that Sum (ar^k) for k = 0 to n = a*Sum(r^k) for k = 0 to n. When in doubt, expand the series for a bit and see what you can do with normal algebraic manipulation.
• What previous video is Sal talking about? The last video didn't cover geometric series.
• Can 'r' be equal to one?
If so, is 2,2,2,2,2 ..... considered a geometric series ?
• Yes, you could classify it as a geometric series.
• Is it possible for us to convert any geometric series to sigma notation? If yes, please tell me the common method to convert it. If no, why can't we do that?
• I'm pretty sure the answer is yes. Say your first term is a and the common ratio is r. The sigma notation would then be Sum from 0 to n of a(r)^n.
• Is every series written in Sigma notation a geometric series?
(1 vote)
• No. All manner of series can be written using ∑ notation.