# Book to learn how to use series expansion intuitively

I'm reading a book written by a (dead) physicist and I'm impressed by the ease with which he uses series expansion to gain intuitive understanding of a function's behavior.

Is there any good book where I can learn this skill?

I'm not talking about the raw mathematics of series expansion (which I know), but rather learning to "read" what the expansion tells, like people did before we relied on computers to graph stuff.

• Which book are you referring to?
– jeea
Apr 15, 2018 at 13:53
• @jeea, "Probability Theory: The logic of Science" by E. T. Jaynes. The book is excellent, I highly recommend it. Apr 20, 2018 at 9:43
• Indeed, a masterpiece.
– jeea
Apr 21, 2018 at 12:48

The "best" textbook I have found so far in this regard is Mathematical Analysis I by Claudio Canuto and Anita Tabacco.

The book builds understanding of functions behavior through the first chapters before tackling Taylor expansions. It contains a lot of illustrations, more than other textbooks I examined.

This is not exactly what I was looking for, but it feels that once enough intuition is built for the behavior of basic functions, the series expansion can better be understood. The chapter about local comparisons helps compare the terms of the expansion.

I am still looking for something more focused, however. Below are the four main features of the book.

The book builds intuition using graphs. Chapter 5 is about local comparison of functions and covers Landau symbols:

• f controlled by g
• f has the same order of magnitude as g
• f equivalent to g
• f negligible with respect to g

It discusses the fundamental limits of sin, cos, log, exp, etc.

And the exercise (with solution) ask to compare several functions relative to each other, which helps build intuition.

In section 6.10 "Qualitative study of a function", functions behavior is studied in a systematic fashion with several examples:

• Domain and symmetries
• Behaviour at the end-points of the domain
• Monotonicity and extrema
• Convexity and inflection points
• Sign of the function and its higher derivatives (inflexion points, etc.)

Numerous exercises with solutions and graphs to go along the function's study. Here is an example : Eventually, Taylor expansions are discussed and illustrated.  