## OPERATIONS ON SERIES:

New series can be generated by making an appropriate substitution in a known series. If you know a common series, and you're given one that resembles it, you can make the appropriate action to find it's expansion and radius of convergence!

## simple substitution - Relation to other series!

If you look at the "x" term in the common series, see if you can discern what the "x" is in the series you have. Simply

**replacing**it in the common series will give you the new one!## Differentiating/integrating

If you

In words....

If f(x) = some function, with f(x) = sum of the terms of the Taylor polynomial….

Then f ’(x) = d/dx (terms of the series that make the function).

~

If f(x) = some function, with f(x) = sum of the terms of the Taylor polynomial….

Then ∫ f(x) = ∫ (terms of the series that make the function).

As you can see, you can manipulate series in many ways!

**differentiate or integrate the function**, that will equal the**sum of**derivatives or integrals of**every term**of the series that created the function.In words....

If f(x) = some function, with f(x) = sum of the terms of the Taylor polynomial….

Then f ’(x) = d/dx (terms of the series that make the function).

~

If f(x) = some function, with f(x) = sum of the terms of the Taylor polynomial….

Then ∫ f(x) = ∫ (terms of the series that make the function).

As you can see, you can manipulate series in many ways!