How do you use differentiation to find a power series representation for #1/(6+x)^2#?
1. Integrate to get a form of
2. Modify the equation to achieve getting precisely
3. Write out the power series with
4. Reverse what you did to re-acquire the original function. i.e. re-multiply by what you divided by (-1 and 1/6, here), then re-differentiate. Whatever returns your original function.
Notice how the power series
Similarly, use -x instead of x. Every odd power is negative, and every even power is positive by virtue of squaring to some order of magnitude (e.g.
x/6 is your new x. Plug it in, use this alternating series from a few lines up, factor in the 1/6 to get back to
Then, re-differentiate the result to get back to