Can anyone recommend any particularly clear, simple, yet thorough sources of explanation of Power Series in general?
I'm a Calculus II student and am a complete beginner with infinite sequences and series, power series, and working with the concept of infinity in this way. I have (or think I have) a solid understanding of limits. I got an A in Calculus I and nearly always get questions regarding limits correct on quizzes and tests.
I feel relatively comfortable with the fundamentals of convergent/divergent series, and most of the tests for convergence and divergence. I'm in need of a lot more practice, definitely, but I'm not confused by this stuff. It's the details of Power Series that seem so vague to me.
In addition to our course textbook and my instructor, Calculus - Early Transcendentals by James Stewart, I've been reading about Power Series in The Calculus Lifesaver, by Adrian Banner, Thomas Calculus by George Thomas, and have watched some of the videos about the topic by Herbert Gross from MIT's Calculus Revisited (who had the most articulate and helpful—for me—explanation I've encountered so far).
All of these sources, clearly, are excellent Calculus resources (except for Stewart's Calculus and my instructor). Each has given me insight into different aspects of the nature of power series and their uses. But still I feel as if I'm missing something.
For instance, I know that power series are used extensively in many fields of science and engineering to approximate values of functions which can't be easily computed directly. But, based on my understanding, for power series to represent a function, you must be able to manipulate the function so it conforms to the pattern of convergence of a geometric series; e.g.,
$$\text{ If } f(x)=\frac{1}{3-x} \text{ then }S_N = \frac{a}{1-r}$$
$$S_N = \frac{1}{3}\left(\frac{1}{1-\frac{x}{3}}\right) \text{ , etc. }$$
Even then, the range of x values for which the resulting series converges can be tightly restricted. So, how is it possible that Power Series can be so extraordinarily useful in approximating functions if the way they are able to represent functions is so tightly constrained? I know I must be missing something here, but am not sure what. And this is just one example of the trouble I'm having wrapping my head around these new concepts.
Thanks in advance! I always find that different perspectives on a problem can make it easier to understand.
Are you familiar with the concept of Analytic Continuation from Complex Analysis?. I guess you may not have studied Complex Analysis yet, but power series enable us to approximate functions and study their behaviour 'near' to singularities. i.e. $\frac{1}{1-x}$ is not defined for $x=1$ but when $|x| < 1 , \frac{1}{1-x} = \sum_{n=1}^{\infty} x^{n}$,this example is the Geometric Series Formula. Rudin's book is a must for beginners in analysis.