Question

what is sigma notation ?

Answer 1

a way to describe how to add tings up

Answer 2

or even things

Answer 3

for example\[\sum_{i=1}^{5}i=1+2+3+4+5\]

Answer 4

or another example\[\sum_{i=1}^{5}i^2=1^2+2^2+3^2+4^2+5^2\]

Answer 5

and you just add all of them up?

Answer 6

yes

Answer 7

as opposed to \[\prod_{n=1}^{5}\frac{1}{n^2} =\frac{1}{1}*\frac{1}{4}*\frac{1}{9}*\frac{1}{16}*\frac{1}{25}\]

Answer 8

and if you didn't know how many things to add you can write n for the general last term, i.e.\[\sum_{i=1}^{n}=1+2+3+...+(n-1)+n\]

Answer 9

that was supposed to be\[\sum_{i=1}^{n}i\]

Answer 10

and it turns out that\[\sum_{i=1}^{n}i=1+2+3+...+(n-1)+n={n(n+1)\over2}\]so that is when series (sigma notation) starts to get cool, when you can make general formulas for various situations

Answer 11

\[\sum_{n=0}^{4} (2n ^{2}+1)\]

Answer 12

how would you work that out ?

Answer 13

well since you are only goin up to 4 just do it manually:\[(2(0)^2+1)+(2(1)^2+1)+(2(2)^2+1)+...\]hopefully you see the pattern
then just add 'em up

Answer 14

ohhh that makes sense now!

Answer 15

but there various ways to do it, this is just the most simple
it won't work (or at least would be a \(huge\) pain in the butt) for larger upper bounds though
like if you had to add this up from n=0 to n=1000 you would need some more techniques, which you will learn in the future

Answer 16

I'm glad it makes sense now, happy to help :)

Answer 17

\[\sum_{n=1}^{6} 3i\]
so for this it wud be 3(1)+3(2)+3(3)+3(4)+3(5)+3(6)
and than you add them up, right ?

Answer 18

you got it :)

Answer 19

now since you're so quick to learn this, notice one thing...

Answer 20

\[3(1)+3(2)+3(3)+3(4)+3(5)+3(6)=3(1+2+3+4+5+6)\]right?
so we have that\[\sum_{n=1}^{6}3n=3\sum_{n=1}^{6}n\]this idea will help us do more complicated series

Answer 21

in general, this leads us to the idea that\[\sum_{i=1}^{n}Cn=C\sum_{i=1}^{n}n=\frac{Cn(n+1)}2\]so now we have a general formula for the sum of n integers times a constant!
I think this is a good first lesson in series manipulation

Answer 22

typo*\[\sum_{i=1}^{n}Ci=C\sum_{i=1}^{n}i=\frac{Cn(n+1)}2\]

Answer 23

now that got me confused

Answer 24

that's okay, just focus on the part you understood and come back to this part later
you now understand the basics of sigma notation, that's pretty good :)

Answer 25

\[\sum_{n=2}^{10}2/n\]
how would you work this out ?

Answer 26

I typed a bunch of stuff and lost it :/
anyway, for now just do the same trick starting at n=2
2/2+2/3+2/4+...
really do read what I wrote above; understanding the rules behind it all will make this a lot easier
I'm out, goodnight!