Find the Best Book on Power Series for Calculus

[modmans2ndcoming]

is ther ea book that...

Is there a book that is dedicated just to power series? That is one of my week points from calc and rather than look through the poor examples in my old textbook, I would like to just work on a book dedicated to power series that has many good and descriptive examples.

Any "power series for dummies" books out there :-D

if not, then maybe one of the profs on here could start a nice series of books that focuses on one aspect of calc an or other areas of math that can dedicate a good amount of time talking about the examples, that way, it makes independent study easy.

 

[mathwonk]

i believe there is a famous book by the great author konrad knopp on power series.

I also like the treatment in the first chapter of the book "complex analysis of one and several variables" by henri cartan, available in paper now cheaply.

an excellent reference, as for everything else in calculus, is the chapter on power series in the book by courant, vol 1.

the place to start is with the geometric series:

here is a primer:

if r is number, then we seek a formula for the

sum 1 + r + r^2 + r^3 +...+r^n = S. The trick is multiply by r and subtract.

I.e. then rS = r + r^2 + r^3 +...+r^n + r^(n+1)

hence S-rS = 1 - r^(n+1), since all other etrms cancel.

Thuas factorinbg gives S(1-r) = 1 - r^(n+1), so S = (1-r^[n+1])/(1-r).


Now if |r| < 1, then r^(n+1) goes to zero as n goes to infinity, so we get a formula for the infinite geometric series:

1 + r + r^2 + r^3 +...+r^n +...

= limit of (1 + r + r^2 + r^3 +...+r^n ) as n goes to infinity

= 1/(1-r).



Next principle is that any series of positive numbers, whose terms are all smaller than those of some convergent series such as a convergent geometric series, is also convergent.

E.g. consider the power series for e^x

= 1 + x + x^2/2! + x^3/3! +...+x^n/n! + ...

given any fixed value of x, if you choose n larger than x, then each term after the nth term is obtained from the rpevious oen by multiplication by a number smaller than r = x/n < 1. hence this series is eventually smaller than a geometric series hence it converges for all x.

Similar considerations lead to the various standard criteria for convergence of a power series and the concept of radius of convergence.

 

[modmans2ndcoming]

thanks for that. a lot of what you said is also similar to what we covered in my Real Number Analysis class last winter.

it is considered a good thing when knowledge you have gained while in school begins to congeal into one mass of information right? at the moment, I am near the end of my undergrad majors in Math and Computer science and for the life of me I cannot recall what class I learned something in (unless it is obvious like integration methods or something)

 

[mathwonk]

yes that is a great sign! good for you.

 

[mathwonk]

let me recall for you a few gems of wisdom from courant, thus confirming that they are so well explained there that I have remembered them for over 40 years.

the basic principle for convergence of series of positive numbers is the comparison principle: if a series of positrive numbers is smaller tern by term than the terms of a known convergenmt series, then the smaller series is also convergent. Thus convergence tests of this type are distinguished by what is the choice of a series to compare them with. The most fundamental series for comparison is the geometric series, and the other fundmental object for comparison is an integral.

For example to prove convergence of the series: summation of 1/x^r where n > 1, we compare with the integral of 1/x^r, which converges by evaluating the corresponding improper integral.

Comparison with geometric series leads to the "ratio test" and the "root test" for convergence. For instance the ratio test says that if the ratios of consecutive terms a(n)/a(n-1) converges to a number r less than 1 then the series converges by comparison with the geometric series.


i really recommend acquiring courant's calc book. there is a 60 page chapter on infinite series, pp.365-424, and the enxt one is on Fourier series. Indeed virtually every calc question anyone has asked on this forum is answered clearly in courant to my best knowledge. If everyone studied from that book we would not have much to say to them here.

the book is available used from dealers on abebooks.com for about $50, and the similar rewrite by courant and john, is available from about $25. courant has an old world charm and readability that has been compromised a bit in the more dry and modernized courant and john, but the latter is also excellent and an even better bargain at half the price. either one is an incredible bargain, given that vastly inferior books which are almost worthless sell today for over $100, and even over $130.

 

[modmans2ndcoming]

Excellent! thanks for the recommendation. I will look it up and (hopefully if the wife allows :-D ) I will buy it.