Taylor Series Interval of COnvergence and Differention + Integration of it

[Square1]

OK...
"A power series can be differentiated or integrated term by term over any interval lying entirely within the interval of convergence"

When i do term by term differentiaion or t-by-t integration of a series though, am i making use of this fact?
Does this come into play later in a class?

Or is this fact referring to if i wanted to take the rate of change of the summation, then i can only do it on the interval of convergence.

Any response or even just an elaboration would be helpfull. Kind of confused about the matter. Thanks.

 

[Mark44]

OK...
"A power series can be differentiated or integrated term by term over any interval lying entirely within the interval of convergence"

When i do term by term differentiaion or t-by-t integration of a series though, am i making use of this fact?

Your result will be valid only if you do so on an interval that is withing the interval of convergence.

Does this come into play later in a class?

Probably later in the class you're in. One of the things you do with series is use a finite number of terms to approximate the whole series. The interval of convergence can help you get an upper bound on how far off your approximation is.

Or is this fact referring to if i wanted to take the rate of change of the summation, then i can only do it on the interval of convergence.

You can differentiate your series term by term, but if you evaluate your derivative at a specific number, that number has to be within the interval of convergence.

Any response or even just an elaboration would be helpfull. Kind of confused about the matter. Thanks.

 

[Vargo]

You could think of it this way. Suppose you have some functions f(t), g(t), h(t), etc. and you have to do some calculation involving operations like addition, multiplication, division, function composition, inversion, differentiation, integration.. Those calculations can be tricky for general functions, but they are easier for polynomials. The point of power series is that they are glorified polynomials and that you can replace f(t), g(t), h(t), etc. with their power series representations and do the calculations with those instead. So this theorem tells you that it is valid to do that with the operations of differentiation and integration.

When you do term by term differentiation of a power series, you are using this fact if you are assuming that your result represents the derivative of function represented by the original power series.