Power+Series+for+Basic+Functions

=Power Series for Basic Functions =



**__What is a Power Series?__** A power series is a representation of a function using sigma notation, usually in the form: They can be both Taylor Series (centered around a non-zero value for x) or Maclaurin Series (centered around zero). The value that the series is centered around is represented above by c, so a series centered at x = 5 //could// be: Power series are used when a function needs to be represented as a polynomial, but is not currently in that form.

__**Partial Sums**__ Partial sums are used to approximate a value of a function using a finite number of terms. To find the n-th partial sum, plug your value of x into the first n terms of the function. To find the 2nd partial sum, take only the first two terms, which could be represented as: The 4th partial sum might be: Partial sums are also called "order n approximation" (will be used later).

**__How to Represent Power Series'__** In a power series, every n-th term is representative of the n-th derivative of a function at x = c. When making approximations, one first uses a linear approximation, then uses the first derivative to get the correct slope, then uses the second derivative to get the correct concavity, etc. For example, one can equate the derivatives of a function in order to find out the value needed in a Power Series:
 * [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=f%28x%29%20%3D%20sin%28x%29%0A]] || [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=f%280%29%20%3D%200]] ||
 * [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=f%27%28x%29%20%3D%20cos%28x%29%0A]] || [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=f%27%280%29%20%3D%201]] ||
 * [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=f%27%27%28x%29%20%3D%20-sin%28x%29%0A]] || [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=f%27%27%280%29%20%3D%200]] ||
 * [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=f%27%27%27%28x%29%20%3D%20-cos%28x%29%0A]] || [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=f%27%27%27%280%29%20%3D%20-1]] ||
 * [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=f%5E%7B%284%29%7D%28x%29%20%3D%20sin%28x%29%0A]] || [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=f%5E%7B%284%29%7D%280%29%20%3D%200]] ||
 * [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=f%5E%7B%285%29%7D%28x%29%20%3D%20cos%28x%29%0A]] || [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=f%5E%7B%285%29%7D%280%29%20%3D%201]] ||


 * [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=c_0%20%3D%200]] ||
 * [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=c_1%20%3D%20%5Cfrac%7B1%7D%7B1%21%7D%20%3D%201]] ||
 * [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=c_2%20%3D%20%5Cfrac%7B0%7D%7B2%21%7D%20%3D%200]] ||
 * [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=c_3%20%3D%20-%5Cfrac%7B1%7D%7B3%21%7D%20%3D%20-%5Cfrac%7B1%7D%7B6%7D]] ||
 * [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=c_4%20%3D%20%5Cfrac%7B0%7D%7B4%21%7D%20%3D%200]] ||
 * [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=c_5%20%3D%20%5Cfrac%7B1%7D%7B5%21%7D%20%3D%20%5Cfrac%7B1%7D%7B120%7D]] ||


 * [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=P%28x%29%20%3D%20x%20-%20%5Cfrac%7B1%7D%7B6%7Dx%5E3%2B%5Cfrac%7B1%7D%7B120%7Dx%5E5%20%2B%20...]] ||

**__6 Common Power Series__** __(1)__ __Power Series:__

__Graphical Representation:__

__(2)__ __Power Series:__

__Graphical Representation:__

__(3)__ __Power Series:__

__Graphical Representation:__

__(4)__ __Power Series:__ (centered around x = 1)

__Graphical Representation:__

__(5)__ __Power Series:__

(No Graphical Representation available)

__(6)__ __Power Series:__

(No Graphical Representation available)

Sources: WolframAlpha (all graphs, series images) Wikipedia (Information, some images)