Pythagorean Identities :There are two types of identitties:

The following information was found at:http://coolmath.com/pythagoreanidentities.htm

Let's start with the Unit Circle:

Zooming in to get a better look, we can label the sides of our right triangle...

By the Pythagorean Theorem, we have that

Since this is a Unit Circle, we have

Substituting, gives us

(The first Pythagorean Identity)

Most people easily remember this first identity... But, often times, you'll need to remember the other two when you get to Calculus.

Here's an easy way to derive the next two, starting from the first one:

2) Divide everywhere by 

Simplifying gives us the second Pythagorean Identity...

3) Divide everywhere by 

Simplifying gives us the third and final Pythagorean Identity...

Theb following Explanatin of Power Series was found at:http://en.wikipedia.org/wiki/Power_series

 Power series :

Any polynomial can be easily expressed as a power series around any center c, albeit one with most coefficients equal to zero. For instance, the polynomial f(x) = x² + 2x + 3 can be written as a power series around the center c = 0 as

or around the center c = 1 as

or indeed around any other center c. One can view power series as being like "polynomials of infinite degree," although power series are not polynomials.

The geometric series formula

which is valid for | x | < 1, is one of the most important examples of a power series, as are the exponential function formula

and the sine formula

valid for all real x. These power series are also examples of Taylor series. However, there exist power series which are not the Taylor series of any function, for instance

Negative powers are not permitted in a power series, for instance is not considered a power series (although it is a Laurent series). Similarly, fractional powers such as x^{1 / 2} are not permitted (but see Puiseux series). The coefficients a_n are not allowed to depend on x, thus for instance:

is not a power series.

[edit] Radius of convergence

A power series will converge for some values of the variable x (at least for x = c) and may diverge for others. There is always a number r with 0 ≤ r ≤ ∞ such that the series converges whenever |x − c| < r and diverges whenever |x − c| > r. The number r is called the radius of convergence of the power series; in general it is given as

or, equivalently,

(see limit superior and limit inferior). A fast way to compute it is

if this limit exists.

The series converges absolutely for |x - c| < r and converges uniformly on every compact subset of {x : |x − c| < r}.

For |x - c| = r, we cannot make any general statement on whether the series converges or diverges. However, Abel's theorem states that the sum of the series is continuous at x if the series converges at x.