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I would be very happy if it covers sequence and series from very basics to advanced. Thanks.:)

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Does it make sense to 'accept' an answer in community wiki? –  user8786 Mar 30 '11 at 6:48
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3 Answers 3

The one book I own which is devoted entirely to sequences and series is Knopp's Theory and Applications of Infinite Series. I wouldn't call it advanced, but it contains slightly more than I've ever needed or wanted to know, so I am very happy with it.

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There is also Infinite Sequences and Series by Knopp, which is a sort of shortened edition of the bigger book suggested above. –  Pacciu Mar 28 '11 at 10:34
    
@Pacciu: hmm, yes, that is vaguely familiar. I will stand by my answer though: the larger book of Knopp only has slightly more than I wanted and is itself quite cheap (being a Dover book), so why not splurge for the extra material? –  Pete L. Clark Mar 28 '11 at 13:41
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Bonar and Khoury's Real Infinite Series is an excellent book on sequences and series of real numbers. There are many examples in the book, but unfortunately no exercises.

Many calculus books have numerous exercises on the basics of sequences and series.

For more challenging exercises and for material on series of functions (like power series and Fourier series), you can try a book like Apostol's Mathematical Analysis or Rudin's Principles of Mathematical Analysis. Bartle's Introduction to Real Analysis, 3rd ed. is also excellent and might be more accessible. Bartle's book also has answers to selected problems. Solutions for the exercises in all three books are available online, if you look hard enough.

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Another book that you could look at would be Infinite Series by Earl David Rainville. I used this book as a reference for some basics about infinite products (which as you can imagine is closely related to infinite series) for my senior thesis. I didn't use the rest, but it looked like a good discussion of infinite series going above what is seen in a Calc II class, including a couple other tests for convergence.

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