Dear MathOverflow community,

In about a year, I think I will be starting my undergraduate studies at a Dutch university. I have decided to study mathematics. I'm not really sure why, but I'm fascinated with this subject. I think William Dunham's book 'Journey through Genius ' has launched this endless fascination.

I can't wait another whole year, however, following the regular school-curriculum and not learning anything like the things Dunham describes in his book. Our mathematics-book at school is a very 'calculus-orientated' one, I think. I don't think it's 'boring', but it's not a lot of fun either, compared to the evalutation of $\zeta(2)$, for example. Which is why I took up a 'job' as as a tutor for younger children to help them pass examinations. I wanted to make money (I've gathered about 300 euros so far) to buy some new math-books. I have already decided to buy the book ' Introductory Mathematics: Algebra and Analysis' which should provide me with some knowledge on the basics of Linear Algebra, Algebra, Set Theory and Sequences and Series. But what should I read next? What books should I buy with this amount of money in order to acquire a firm mathematical basis? And in what order? (The money isn't that much of a problem, though, I think my father will provide me with some extra money if I can convince him it's a really good book). Should I buy separate books on Linear Algebra, Algebra and a calculus book, like most university web-pages suggest their future students to buy?

Notice that it's important for me that the books are self-contained, i.e. they should be good self-study books. I don't mind problems in the books, either, as long as the books contain (at least a reasonable portion) of the answers (or a website where I can look some answers up).

I'm not asking for the *quickest* way to be able to acquire mathematical knowledge at (graduate)-university level, but the *best* way, as Terence Tao once commented (on his blog): "Mathematics is not a sprint, but a marathon".

Last but not least I'd like to add that I'm especially interested in infinite series. A lot of people have recommended me Hardy's book '*Divergent Series*' (because of the questions I ask) but I don't think I posess the necessary prerequisite knowledge to be able to understand its content. I'd like to understand it, however!

`_`

or asterisks`*`

on either side, not dollar signs. In TeX, use`{\em`

text`}`

. Dollar signs make the computer process whatever's inside as math, as if you had all those variables to multiply together. The classic example is $difference$ versusdifference— notice the spacing around the`f`

s. (In the default TeX font, the correct look is`$\textit{difference}$`

.) The spacing is even weirder for words with`ffi`

:`$spiffier$`

,`$\textit{spiffier}$`

,spiffier. – Theo Johnson-Freyd Jun 14 '10 at 22:10