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AP®︎ Calculus AB (2017 edition)

Unit 8: Lesson 4

Summation notation

Summation notation

Sigma, Σ, is the standard notation for writing long sums. Learn how it is used in this video. Created by Sal Khan.

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• At , shouldn't the i to the right of the sigma be i+1?
• No, it has to be i. We're adding the numbers from 1 to 10. The sigma notation says we're going to add up the results of applying the rule to the right of the symbol to each of the i's from 1 to ten. In this case we're applying a rule that does nothing, just gives back i. So the first result in the addition is 1, then 2 and so on up to 10. If we had i+1 to the right of the symbol, the first result in the addition would be 2, and we would end up adding the numbers from 2 to 11.
• So, is sigma notation is kind of like a for loop in programming?
• As a while loop in python:
import math
i = 0 #starting index
n = 0 #result of that index number
s = 0 #total sum
while i <= 50: #ending index
n = math.pi*i**2
print(str(i) + "|" + str(n))
i += 1
s = s + n
print("sum: " + str(s))
• what happens if it is constant summation
• You would simply multiply the constant by the number of terms. For example: if the constant is 8 (Ʃ8), and there are six terms, you would just multiply 8 and 6, to get 48. I hope that helps!
• At the end of the video, I'm just wondering could the index be a decimal? If so, what if the number on top of the sigma was a integer? How would that work? I'm just curious and brand new to sigma. Tell me if my question is completely unreasonable.
• No, the sum notation does not accept negative number, fraction or decimal.
• So by default, as in when you don't put anything else in front of the i next to the Greek letter, whatever i is set as equal to below the letter increases by 1 in the summation? So that when you want to set the summation as going 1+2+3+4+5+6+7+8+9+10 you just write down i next to the Greek letter, then when you put, say, 2 next to the i as in 2i, the summation goes 2+4+6+8+10+12+14+16+18+20 and not 2+2+2+2+2+2+2+2+2+2?
• Correct; also, I think you mean Sigma when you are referring to the "Greek letter".
• Can we set the index as a negative number? Also, can the upper boundary be a negative number?
• The definition of the sigma notation seems to be surprisingly vague, and I couldn't find anything that would explicitly prohibit the usage of negative indices or negative boundaries, as long as they are used in a consistent way (lower boundary < upper boundary).
However, I would rather avoid doing so for several reasons:

1) The sigma notation basically represents the terms of a series, and each term is usually associated with a letter and the corresponding index (e.g. a1 for the first term, a2 for the second one, and so on). It would be more than unconventional to use negative indices for these terms (e.g. a-5).

2) I've never seen the usage of negative indices or boundaries in any text book. Since the sigma notation is basically just a convention to write out long sums in a short way, it's probably best to stick to the prevailing convention of using non-negative indices and boundaries only.

Summarized, I wouldn't go as far as calling the usage of negative indices and boundaries as 'wrong', but it will at least raise some eye brows.
• Is there a video out there with a more in depth explanation on how to use summation notation, especially with integrals? Unfortunately, the first time I ever saw summation notation was in the heat of integral calculus a few weeks ago. I am finding it extremely difficult to understand all of the different symbols and how they are being used...
• If I wanted the increment to be more than 1, would there be a specific notation for that, or would I just have to implement it into the right - hand side of the Sigma notation?
• Increment augmentation is done to the right of the Summation function.