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Tuesday, 12 September 2017

3.1 Sequence and Series

A SERIES is the sum of a sequence.
Here's a sequence:
1 , 2 , 3 , 4 , 5
Here's the corresponding series:
1 + 2 + 3 + 4 + 5
We have a special notation for series.
First, let's get the formula for the nth term of the above sequence...
an = n

sigma
It's an "S" in the Greek alphabet.
Think of it as an "S" for "sum!"

EXAMPLE:
Our series adds five terms:
1 + 2 + 3 + 4 + 5

the summation of k as k goes from 1 to 5

the summation of k as k goes from 1 to 5 ... sigma for summation, k is the formula for the nth term, k is the index (it's like a counter. some books use i.) , 1 is the term we start with , and 5 is the term we end with
* Notice that we're using a k instead of the n...
This is important and will make something easier later.

So, for this sequence whose nth term is given byan = n, we
have  
the summation of k as k goes from 1 to 5 ... start at k = 1, which gives 1, then let k = 2, which gives + 2, then let k = 3, which gives + 3, then let k = 4, which gives + 4, Finally, let k = 5, which gives + 5. Done! ... add them up! = 15

Let's find the sum:
the summation of 2k as k goes from 3 to 6
the summation of 2k as k goes from 3 to 6 ... k = 3 gives 2 ( 3 ) , k = 4 gives 2 ( 4 ) , k = 5 gives 2 ( 5 ) , k = 6 gives 2 ( 6 ) ... add them up ... = 36


THE EVENS:
the summation of 2k as k goes from 1 to infinity will generate 2 + 4 + 6 + 8 + ...
the summation of 2k as k goes from 1 to infinity ... notice the infinity
This means the series goes on forever and ever.
If you want to generate
0 + 2 + 4 + 6 + 8 + ...
what would you need to change?
the summation of 2k as k goes from 0 to infinity ... start the index at 0!

THE ODDS:
Odd numbers are just evens plus one...
the summation of ( 2k + 1 ) as k goes from 0 to infinity = 1 + 3 + 5 + 7 + ...
Or you can think of odd numbers as evens minus one...
the summation of ( 2k - 1 ) as k goes from 1 to infinity = 1 + 3 + 5 + 7 + ...

the summation of ( 2k + 1 ) as k goes from 0 to infinity and the summation of ( 2k - 1 ) as k goes from 1 to infinity
Here's what happened...              
These guys started at different places.
So, be careful and ALWAYS check the first few terms to make sure that everything works!


ALTERNATING SIGNS:
These will come up a lot!
We know how to generate the evens
2 + 4 + 6 + 8 + 10 + ...
But, what if the signs alternate?
2 - 4 + 6 - 8 + 10 - ...
Using one of these will fix it:
( -1 )^k or ( -1 )^( k + 1 ) or ( -1 )^( k - 1 )
Which of these you use depends on where you start your index and if the thing starts with a positive or a negative.  
the summation of 2k ( - 1 )^k as k goes goes from 1 to infinity = -2 + 4 - 6 + 8 - ... (the ( -1 ) is multiplied in)
the summation of 2k ( -1 )^( k + 1 ) as k goes from 1 to infinity = 2 - 4 + 6 - 8 + ...
the summation of 2k ( -1 )^( k - 1 ) as k goes from 1 to infinity = ___


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