What are Series?

Series are sequences added up. Sequences are a list of numbers in order. For examepl, the sequence `color(black)(a_n=a_1+(n-1)d)` is the equation for a arithmetic sequence. If `color(black)(a_1=3, d=2)`, then the sequence would be {3, 5, 7, 9...}. If you added up all those numbers in the sequence, then you would have a series.

Notation for the series of the arithmetic sequence would be `color(black)(sum_(n=1)^oo 3+2(n-1))`

This series would be divergent because it would go to infinity. If a series equals a certain number, it is convergent.

Series Tests

Using these tests will tell you whether or not a series is convergent or divergent.

Conditional and Absolute Convergence

When you find an alternating series that converges, it only conditionally converges until you determine that it absolutely converges. Absolute convergence means that the series will also converge even after you take the absolute value of the series. If the series does not absolutely converge, it can still conditonally converge or diverge.

For example, `color(black)(sum_(n=1)^oo(-1)^n 1/n)` conditionally converges due to the alternating series test because the series 1/n exists for all n > 1, it is decreasing, and it approaches 0. It is not absolutely converge because `color(black)(sum_(n=1)^ooabs((-1)^n 1/n)=sum_(n=1)^oo 1/n)` diverges due to p-series because it is `color(black)(1/(n^1))` and according to p-series it diverges.

Alternating Series Error Bound

If you have an alternating series and you find the Nth partial sum of a converging alternating series, then `color(black)(abs(S_oo - S_N) <= abs(a_(N+1)))`

Power Series

Power series is an infinite series that is an exact representation of an elementary function.

Power series take the form of `color(black)(sum_(n=1)^ooa_nx^n)` or `color(black)(sum_(n=1)^ooa_n(x-c)^n)`

Power series can either converge at one point, for all x, or only over an interval.

If you are given the power series `color(black)(sum_(n=1)^ooa_n(x-c)^n)` then this means that the series either converges only at x=c, or for all x, or on an interval about c with radius R so that if `color(black)(abs(x-c) < R)` then the series converges absolutely. If `color(black)(abs(x-c) > R)` then the series diverges.

Properties of Functions Defined by Power Series

If a function `color(black)(f(x) = sum_(n=1)^ooa_n(x-c)^n)` has a radius of convergence of R > 0, then on the interval (c-R, c+R) f is differentiable and also continuous. In that case, `color(black)(f'(x) = sum_(n=1)^oona_n(x-c)^(n-1))` or `color(black)(intf(x)dx = c + sum_(n=1)^oo(a_n(x-c)^(n+1))/(n+1))`

Something to note is that when differentiating or integrating, the radius of convergence of the series obtained by doing such operations is the same as that of the original power series. However, the interval of convergence may be different.

Operations with Power Series

If `color(black)(f(x)=sum_(n=0)^ooa_nx^n)` and `color(black)(f(x)=sum_(n=0)^oob_nx^n)` then:

1. `color(black)(f(kx)=sum_(n=0)^ooa_nk^nx^n)`

2. `color(black)(f(x^N)=sum_(n=0)^ooa_nx^(Nn))`

3. `color(black)(f(x)+-g(x)=sum_(n=0)^oo(a_n+-b_n)x^n)`

Radius of Convergence

You can find the radius of convergence through the use of the ratio test, because you will end up with the form `color(black)(abs(x-c)` > 1. After you find your interval of convergence, you must test the end points to see if the series is convergent at the end points.

Geometric Power Series

The sum of a geometric series is written as `color(black)(Sum =a/(1-r)`. If you are given something in the form of `color(black)(a/(1-r)` then you can rewrite it in a the form of a power series as `color(black)(sum_(n=0)^ooar^n)`.

For example, if you are given the equation `color(black)(3/(2+x^2)` which is equal to `color(black)(3/(2(1-(-1/2x^2)))` which equals `color(black)((3/2)/((1-(-1/2x^2))))`. When you put this in the form of a power series you will get `color(black)(sum_(n=0)^oo(3/2)(-1)^n(1/2x)^(2n))`.

Taylor Series

Taylor series is a power series that represents a function centered around the x=c. Taylor series are generally in the form of

`color(black)(sum_(n=0)^oo f^(n)(c) (x-c)^n/(n!))` `color(black)(=f(c)+(f'(c)(x-c))/(1!)+(f''(c)(x-c)^2)/(2!)+...)`

You can see that the Taylor series representation becomes more accurate as you have more terms. If you click on edit graph in the graph below and slide the a= slider, you can see how the representation becomes more accurate.

Lagrange Error Bound

If `color(black)(P_n(x))` is the nth degree Taylor polynomial of f(x) with a center at x=c then the error will be

`color(black)(abs(f(x)-P_n(x)) <= M/((n+1)!)(abs(x-c))^(n+1))`

Where `color(black)(abs(f^(n+1)(a))<=M)` or maximum for all a between x and c.

MacLaurin Series

MacLaurin Series are power series that represents a function centered around x=0, so they are in the form of

`color(black)(sum_(n=0)^oo f^(n)(0) (x)^n/(n!))` `color(black)(=f(0)+(f'(0)(x))/(1!)+(f''(0)(x)^2)/(2!)+...)`

Here are some MacLaurin series representation of functions (centered around 0) that you will need to know for the AP test.

`color(black)(e^x=sum_(n=0)^oo(x^n)/(n!))`

`color(black)(ln(x+1)=sum_(n=1)^oo(-1)^(n-1)(x^n)/(n))`

`color(black)(sin(x)=sum_(n=1)^oo(-1)^(n-1)(x^(2n-1))/((2n-1)!))`

`color(black)(cos(x)=sum_(n=0)^oo(-1)^(n)(x^(2n))/((2n)!))`