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.

  • $\begingroup$ Rudin's PMA has a section on power series, and it's a very valuable book on analysis. If you want something more complete, you may have a look at the first chapters of a complex analysis book. Henrici's "Applied and Computational Complex Analysis" starts with power series, is very clear, and has lots of applications. $\endgroup$ – Jean-Claude Arbaut Nov 12 '15 at 0:09
  • $\begingroup$ Except in certain special (but important) cases, Taylor polynomials are much more useful than Taylor series, for exactly the reasons to which you are alluding: Taylor series frequently do not converge, and frequently even when they do converge, they converge to a different function than the one you want. $\endgroup$ – Ian Nov 12 '15 at 0:12
  • $\begingroup$ @Jean-ClaudeArbaut Thank you very much. I will check those out. I have a copy of PMA on loan from the library. Many people have suggested turning to classic books on Analysis for clarifications of Calculus topics. I also have a copy of "Introduction to Calculus and Analysis" by Courant. I've read several chapters, about derivatives, integrals and series. But I was surprised many more recommendations weren't for more recent books. Does the arcane language and the use of Greek letters for variables and functions make them more challenging to understand? $\endgroup$ – tommytwoeyes Nov 13 '15 at 22:53
  • $\begingroup$ @Ian Thank you for pointing this out. Nice to know I'm not completely on the wrong track. I think part of my difficulty is that I'm having trouble visualizing how the pieces--power series, Taylor series, Maclaurin series, and others--fit into the landscape of the problems they were invented to solve. I've thought about investigating the historical accounts of how those tools were developed, but my time is already consumed with my coursework. I've even searched for UML-type diagrams or mind maps that might illustrate the relationships between those tools, but I haven't found anything. $\endgroup$ – tommytwoeyes Nov 13 '15 at 22:57
  • $\begingroup$ @tommytwoeyes Taylor polynomials achieve local approximation of relatively smooth functions. Taylor series give you a simple global formula for certain nice functions, including polynomials, exponentials, sine, cosine, etc. Yet they are also incredibly limited. For instance they do not even work globally for rational functions which have no singularities. There is a famous example, $\frac{1}{x^2+1}$, which is a completely smooth function--but its Maclaurin series is divergent for $|x| \geq 1$. This fact is closely related to some of the comments that have been made about complex variables. $\endgroup$ – Ian Nov 13 '15 at 23:33

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.

  • $\begingroup$ Sean: No, this is the first I've heard of Analytic Continuation. Are you saying that power series help in the study of function behavior near singularities because the individual terms allow you to sort of "zoom in" on the behavior of the function the series approximates at points near to the asymptote? On a side note, so many people have pointed me to Analysis for help with Calculus, I've wondered why we don't just replace Calculus with Analysis in education. Most schools have 2-3 levels of Algebra. Why not Analysis I-VI? Why is there a distinction between Calculus and Analysis? $\endgroup$ – tommytwoeyes Nov 13 '15 at 23:04

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