Why can no one explain Power Series and Functions clearly

[JoeTheKid]

Hello,
Im currently in a Calc II class with unfortunately a bad professor (score of 2 on RateMyProfessor), so I often have to resort to outside sources to learn. Our class is currently on Sequences and Series which has been fine up until we hit the topic of relating Power Series and Functions.

Example: ∑ x^n = 1/(1-x) when |x|<1

Now we receive weekly homework assignments, our prof went over differentiation and integration of power series VAGUELY with a few examples that don't help. So naturally I turned to the internet for help, whilst going through source after source that apparently is explaining this stuff, I can comfortably say that I have no idea what is going on in problems such as this.

f(x) = ∑((1)/((4^n)(n^2))(x-1)^n

x
Find ∫ f(t)dt As a series. Then find the Interval of Convergence
1

I actually don't even know where to start, so if anyone can offer any sort of insight into these types of problems I would be grateful.

 

[Geofleur]

Did you mean "Find $\int_1^x f(t) dt$ as a series" ? If so, why not re-express $f(x)$ as a function of $t$ and stick it into the integral? You could integrate term by term after that.

 

[mathwonk]

the point is that the theory is somewhat hard but the practice is easier. I.e. proving that a convergent series defines a differentiable function (and with the same radius of convergence) takes a bit of work, and the same for integrating, but after knowing that, you just differentiate and integrate them term by term.

e.g. suppose you want a formula for π. just start from 1/(1+x^2), expand by the geometric series you just used, then integrate term by term to geta series for mula for arctan(x), then plug in x=1, to get formula for π/4. this is really cool. (of course you need to know the series makes sense at this point on the edge of the circle of convergence, but so what?)