14-3: Series and Partial Sums

Okay, I'm sorry if this is really confusing, but I didn't really get it today in class. (ESPECIALLY not the binomial part)

The difference between a series and a sequence:
A sequence is just a list of numbers like 2,4,6,8
A series is different because it's all being added together in a string like 2+4+6+8.

Mr. Frost taught us that to find the sum of a series, you can pair up the different numbers in the series. So if you had the series 3,7,11,15,19,23,27,31,35,39 you could pair it up like:
3 + 39 = 42
7 + 35 = 42
11 + 31 = 42
15 + 27 = 42
19 + 23 = 42

42 x 5 = 210 (the complete sum for the series)
The equation for the partial sum is

S(n) = (t(1) + t(n))(n/2) -- This is for an ARITHMATIC series

The geometric partial sum equation is:

S(n) = t(1) x (1-r^n)/(1-r)

If the absolute value of r is less than 1, however, the n value will only get bigger and bigger, and this will only make the r^n value in the formula smaller and smaller until it eventually doesn't even matter anymore. So, if the absolute value of r is less than 1, the formula for the geometric partial sum is:

S(n) = t(1)/(1-r)

A geometric series will converge to a limit if the common ratio r satisfies the inequatlity "the absolute value of r is less than 1". If the absolute value of r is greater or equal to 1, the series diverges.

Binomial Series
-coefficients of a binomial series can be calculated recursively using the term before it.

(coefficient of a)(exponent of a)/(term number) = coefficient of next term

-coefficients can also be calculuated algebraically. They are equal to the numbers of combinations of n objects taken r at a time. Like in the probability section:

5C2 = 5!/3!2! = 5x4x3x2x1/(3x2x1)(2x1) = 10, so the term containing b^2 is
5!/3!2!a^3b^2

When expanding a binomial (a + b)^n the term with b^r is
(n!/(n-r)!r!)a^(n-r)b^r = (n/r)a^(n-r)b^r = nCra^(n-r)b^r

Practice Problem:
Using the series 7+12+17+22+27+32, find the 6th partial sum.
7 + 32 = 39
12 + 27 = 39
17 + 22 = 39

39 x 3 = 117 you can do it this way OR you can do it the equation way

S(n) = (t(1) + t(n))(n/2)
S(6) = (7 + 32)(3)
S(6) = (39)(3) - which is the same thing from before!
S(6) = 117

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umm... i hope this posted. it's all just html in this format. This is a picture of one of Picasso's most famous blue period paintings. Also my favorite of his blue period.