ABSTRACT
The ninth chapter dives into properties of sequences and sums. It starts from finite arithmetic and geometric sequences and proves theorems for computing their sums. Then it introduces infinite sequences and the notion of a limit. The beautiful Bolzano-Weierstrass theorem is a proof of existence in its purest form—that is, without explicitly pointing to the object in question! Separate sections talk about sequences with nested radicals, including those magically converging to pi. Another section proves the existence of the limit for a sequence that defines the base of the natural logarithms. A discussion of the harmonic sequence is followed by Euler's creative proof that there are an infinite number of primes.
