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## Calculus, all content (2017 edition)

### Course: Calculus, all content (2017 edition)>Unit 7

Lesson 3: Series review

# Converting explicit series terms to summation notation (n ≥ 2)

Usually, when writing a series in sigma notation, we start summing at n=1 or n=0. Sometimes though, we would like to start at n=2 or larger values. See an example here. Created by Sal Khan.

## Want to join the conversation?

• I came up with the formula (-1)^(n-1) * 8(n-1)/(2n+1). Is this correct as well? • Hey, so I just figured out that the common ratio between these successive terms is -10/7, therefore shouldn't the fourth term be -160/49 instead of -32/9?
(1 vote) • Well... In a word, no. The ratio between the first two terms may be -10/7, but the fourth term is part of the series as Sal has defined it, too, so we can't just discount it or say that it must be wrong when we're trying to figure out the pattern. This is not a geometric series, but if you just look at the first two terms, you might think it is. In fact, if you just look at the first two terms of any series, you could convince yourself that it's geometric because there will always be some constant ratio between two given numbers.

If we knew that this series was geometric, your statement would be correct; however, we don't know that, and in fact it's not a geometric series, so we can't assume that there will be a common ratio. Hope that makes sense.
• what does a sub n mean? • a sub n, which people often format as a_n is the last number in a series, or any particular number represented by its position n, in the a series (a_1, a_2, a_3, ..., a_n). If we have 10 numbers in some series, the tenth number, a_10, will be the a_n. In the example in this video, the first term is `a_2`. And `a_n` would be an infinite unkwown number. Bytheway, there are sums that the infinite sum of n numbers is known, like `sum of 2^-n, n=0 to infinity`. http://www.wolframalpha.com/input/?i=sum+of+2%5E-n,+n%3D0+to+infinity
(1 vote)
• How can I identify patterns in series easier? I know it's broad, but that's my main problem when completing these questions. Once I identify the pattern, I can come up with the general term pretty easy. • Look for the usual suspects: repetition, distance between terms, ratio between terms, alternation patterns. The best method is just to practice it. Look for other resources online, calculus texts, etc. Remember way back when factoring a quadratic seemed difficult, but now, with lots of experience under your belt, no problems, right? Same deal here.
(1 vote)
• i have a different formula! is this right [(-2)^n * 2] / -(2n +1) • Can i write it as 4*(-2)^n / 3+2n instead?
(1 vote) • There are an infinite number of ways to express most sequences/series explicitly. Your expression is close; however, at n = 1 your expression = -8/5. If you look at Sal's expression you will notice his series starts off with n = 2. Thus, if you were to decrease the index of your expression by one you would get a correct answer as well. For example,
4*(-2)^(n-1) / 3+2(n-1)
4*(-2)^(n-1) / 3+2n-2
4*(-2)^(n-1) / 2n+ 1

Notice, if you manipulated the final expression above enough, you would get same answer as Sal.

(Focusing only on the numerator)
4 = 2*2
(-2)^(n-1) = (-2)^n / -2
2*2*(-2)^n / -2
2 and -2 cancel out to equal -1
-1*2*(-2)^n
-1 * 2 = -2
-2*(-2)^n = (-2)^(n+1)
Thus, your expression = (-2)^(n+1) / 2n+1
• Could another Sigma notation for this same problem be: (-5)^n/3n?
My reasoning is, (-5)^1 = -5, ^2 = 25, ^3 = -125, etc, and it would be much simpler. Would that work as a solution for this as well? Thanks! Sorry, I'm watching the video "Writing a series in Sigma notation" but the comments are from a different video. Not sure what's going on with the site, but I went back and checked other videos and the same thing is happening there. I'll try to put the comment on the right video.
(1 vote) • Yes, that would work; I guess you couldn't see the video comments, but that idea was at the top. And actually, you could even go so far as to express it as (-5/3)^n, which I think may be even more useful because it really shows the geometric nature of this series. As another user said, I suspect Sal used the (-1)^n notation because he wanted to use the concept of an oscillator, which is a good thing to be able to recognize and apply.   