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## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB > Unit 6

Lesson 3: Riemann sums, summation notation, and definite integral notation- Summation notation
- Summation notation
- Worked examples: Summation notation
- Summation notation
- Riemann sums in summation notation
- Riemann sums in summation notation
- Worked example: Riemann sums in summation notation
- Riemann sums in summation notation
- Definite integral as the limit of a Riemann sum
- Definite integral as the limit of a Riemann sum
- Worked example: Rewriting definite integral as limit of Riemann sum
- Worked example: Rewriting limit of Riemann sum as definite integral
- Definite integral as the limit of a Riemann sum

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# Summation notation

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

## Want to join the conversation?

- At1:26, shouldn't the i to the right of the sigma be i+1?(23 votes)
- 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.(117 votes)

- 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.(44 votes)
- No, the sum notation does not accept negative number, fraction or decimal.(67 votes)

- So, is sigma notation is kind of like a for loop in programming?(51 votes)
- 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))(5 votes)

- what happens if it is constant summation(24 votes)
- 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!(44 votes)

- 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?(12 votes)
- Correct; also, I think you mean Sigma when you are referring to the "Greek letter".(11 votes)

- Can we set the index as a negative number? Also, can the upper boundary be a negative number?(8 votes)
- 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.(17 votes)

- 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...(7 votes)
- A summation has 4 key parts: the upper bound (the highest value the index variable will reach), index variable (variable that will change in each term of the summation), the lower bound (lowest value of the index value - the one it starts at), and an expression.

You can watch videos on summation notation here:

https://www.google.com/url?q=https://www.khanacademy.org/math/integral-calculus/sequences_series_approx_calc/calculus-series/v/writing-series-sigma-notation&sa=U&ved=0ahUKEwiUubeX07PJAhVI2R4KHU7FCUUQFggJMAI&client=internal-uds-cse&usg=AFQjCNFNsts-PWgM_eE2uQ4EqZ9aq4a4FQ

Other than that, the video you asked this question on is probably the best you can go. When I first learned summations, the Wikipedia page helped me a lot so you can try that out as well.(8 votes)

- 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?(5 votes)
- Increment augmentation is done to the right of the Summation function.(5 votes)

- How do you express sigma notation when typing?(5 votes)
- There's mutiple ways. You can use the sigma symbol Σ, you can use latex command \sum or you can write it out in text.(5 votes)

- What about pi notation, why is sigma notation so much moer popular, just curious (pi notation is a capital pi and is used for sequences of constant mutiplying, instead of adding like sigma is)? Thx(3 votes)
- Only because it seems that we add things together more often than we multiply them together. When you need to multiply a list of numbers together, pi notation is a great way to show it!(6 votes)

## Video transcript

What I want to do in this
video is introduce you to the idea of Sigma notation,
which will be used extensively through your
mathematical career. So let's just say you wanted
to find a sum of some terms, and these terms have a pattern. So let's say you want to
find the sum of the first 10 numbers. So you could say 1
plus 2 plus 3 plus, and you go all the
way to plus 9 plus 10. And I clearly could have even
written this whole thing out, but you can imagine it becomes
a lot harder if you wanted to find the sum of
the first 100 numbers. So that would be 1
plus 2 plus 3 plus, and you would go all
the way to 99 plus 100. So mathematicians said, well,
let's find some notation, instead of having to do this
dot dot dot thing-- which you will see sometimes
done-- so that we can more cleanly express
these types of sums. And that's where Sigma
notation comes from. So this sum up here, right
over here, this first one, it could be
represented as Sigma. Use a capital Sigma, this
Greek letter right over here. And what you do is
you define an index. And you could start your
index at some value. So let's say your
index starts at 1. I'll just use i for index. So let's say that i starts at
1, and I'm going to go to 10. So i starts at 1,
and it goes to 10. And I'm going to sum up the i's. So how does this translate
into this right over here? Well, what you do is you
start wherever the index is. If the index is at
1, set i equal to 1. Write the 1 down, and then
you increment the index. And so i will then
be equal to 2. i is 2. Put the 2 down. And you're summing each
of these terms as you go. And you go all the way
until i is equal to 10. So given what I just
told you, I encourage you to pause this video and
write the Sigma notation for this sum right over here. Assuming you've
given a go at it, well, this would be the sum. The first term,
well, it might be easy to just say we'll
start at i equals 1 again. But now we're not going to
stop until i equals 100, and we're going to
sum up all of the i's. Let's do another example. Let's imagine the sum from
i equals 0 to 50 of-- I don't know, let me
say-- pi i squared. What would this sum look like? And once again, I encourage
you to pause the video and write it out,
expand out this sum. Well, let's just
go step by step. When i equals 0, this will
be pi times 0 squared. And that's clearly 0,
but I'll write it out. pi times 0 squared. Then we increase our i. And, well, we make sure
that we haven't hit this, that our i isn't already
this top boundary right over here
or this top value. So now we said i
equals 1, pi times 1 squared-- so plus
pi times 1 squared. Well, is 1 our top value right
over here, where we stop? No. So we keep going. So then we go i
equals 2, pi times 2 squared-- so plus
pi times 2 squared. I think you see
the pattern here. And we're just going to
keep going all the way until, at some point-- we're
going to keeping incrementing our i. i is going to be 49. So it's going to be
pi times 49 squared. And then finally we increment
i. i equal becomes 50, and so we're going to have
plus pi times 50 squared. And then we say,
OK, our i is finally equal to this top boundary,
and now we can stop. And so you can
see this notation, this Sigma notation for this
sum was a much cleaner way, a much purer way,
of representing this than having to write
out the entire sum. But you'll see people switch
back and forth between the two.