Writing+series+from+known+series

**WRITING SERIES FROM KNOWN SERIES** A list of common known series

Writing a series from a known series involves taking a series that is already known and manipulating it to represent a new function

__Simple Substitution__ Sometimes, it's impossible for us to integrate something by hand or we are asked to find the series for a function that looks similar to one we already know, but is slightly different. Instead of deriving the function multiple times like we would do normally to write a new series, most of the time it's easier to manipulate a known series for a function to get the series for a new, but similar function. For example, it would be far easier to find the series for (as you will see below) by manipulating the series for  rather than deriving and finding values for each term of.

The known series is: Plug (2x) in for all (x). Which can be rewritten as: We already know what the series expansion is. Plug in the series found for to get... (Simplify) Which can be rewritten as:
 * Ex. 1**
 * Ex. 2**

__Integration/Derivatives__ Sometimes you are asked to find the power series for a derivative or integral of a function because you can't integrate or derive normally.

It is impossible to integrate this regularly, so a series representation is the most useful option. The known series for is: Plug in for x Which can be rewritten as: Now that we know the series for, we can integrate each individual term to find the integral of the series, which turns into...
 * Ex. 3 - Integrals**

Then plug in 1 and 0 as if you had just integrated. Plug in and solve (calculator) Compare this to the actual integral: Notice how closely our 4th partial sum using a power series approximates the actual value of the integral.

The graph for the integral would look like:

Other times you are asked to find the derivative of a series. Since we already know what the series for is, we're just going to use that function again and find the derivative of the series. The process is very similar to that of when we integrated the series, except obviously instead of integrating each term, we derive each term. From last time we know that the series for is Now, we derive each term. Which can be rewritten as:
 * Ex. 4 - Derivatives**

__Sources Cited__ WolframAlpha LaTeX Equation Editor Wikipedia