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## Riemann sums, summation notation, and definite integral notation

Current time:0:00Total duration:4:27

# Summation notation

## 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.

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