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 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).
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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!
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!