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I've taken up self-study of math and i start using the book called :

Mathematical Analysis I Authors: Canuto, Claudio, Tabacco, Anita

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I would like to start from zero to understand taylor expansion and i'm looking about some textbook with homework problems with step-by-step math answers if possible and cover specialy those subjects:

  • Local comparison of functions.
  • Taylor expansions and applications

I would appreciate any book recommendations.

Thanks in advance.

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  • $\begingroup$ What languages can you read? Also, do you want a treatment of Taylor expansions at the level of an initial calculus course, or for those who are already comfortable with topology, normed vector spaces, etc.? $\endgroup$ – David Jan 27 '16 at 5:17
  • $\begingroup$ English French, since i'm start from zero i prefer Taylor expansions at the level of an initial calculus course. $\endgroup$ – Educ Jan 27 '16 at 5:20
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    $\begingroup$ There is The Taylor Series by Paul Dienes, which can be found in most university libraries, but I believe this is mainly a complex analysis book. $\endgroup$ – Dave L. Renfro Jul 8 '16 at 21:05
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It is strangely difficult to find good introductory analysis books that are also good "calculus" books.

I have the first edition of the book: It is very rigorous, and many points that are usually glossed over are explained very well. For example, the variable substitution method of calculating limits (which is usually not explained at all) is given a theoretical justification, and the subsequent calculation examples are explicitly related to the relevant theorem. The only problem with my edition is that the English translation is ludicrously awful. (On the other hand the ridiculous grammar and idiosyncratic expressions provide a few laughs.) The translator is obviously not a native-speaker of English. But who cares? It's mathematics!

The book is very helpful, if we must find her only flaw is precisely the translation. (I hope that the second edition has improved)

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  • $\begingroup$ thank you for your interpretation. $\endgroup$ – Educ Jan 26 '16 at 12:00
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Would you consider on-line resources like Wikibooks?

I don't recall a book entirely devoted to Taylor expansion, so I suggest you find a calculus book on the topic and among the best books I've ever read about analysis and calculus there are the classics by Apostol and Rudin.

When you're facing your first problems and/or questions, take a look at this: basically it's a collection of good problems in real analysis.

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  • $\begingroup$ Even those are good book but they didn't covers Taylor expansions with details. Thank you anyway! +1 $\endgroup$ – Educ Jan 26 '16 at 11:59
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For Taylor expansions treated at the level of an introductory calculus course, I would recommend either

  • S. Lang, A First Course in Calculus (Chapters 13 and 14 if you use the 5th edition); or

  • T. Apostol, Calculus (Vol. 1, Chapters 7, 11, for single-variable calculus, and Vol. 2, Chapter 9, for the multivariable formula up to the second order).

At a higher level, you could consider Vol. 3 of the book by Ramis, Odoux and Deschamps. If that is too difficult, I imagine that the Cours de mathématiques supérieures series by Doneddu ought to bridge the gap between an introductory treatment and what you'll find in Ramis, though I have no personal experience with Doneddu's books.

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  • $\begingroup$ Thank you i'll check that and give you feedback and asked you if i need more titles. +1 $\endgroup$ – Educ Jan 27 '16 at 5:41
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    $\begingroup$ Sure. ${ \frac{}{} }$ $\endgroup$ – David Jan 27 '16 at 5:48

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