This article you should read first, if you don't have a topic of interest already. First article

An end to the mathematical frustration of man comes with anaylsis of the binomial (1620), as the series solution of the hypotenuse function. Ancient functions

The oldest series, being described in Euclid's Elements, has a wide range of applicability. Geometric Series

Exponents are a simple algebraic construction and notation; they are used in scientific notation and are implicit in the decimal notation. Roots are an intrinsic part of exponents (the fractional exponents). So, in this section the "original" exponent value (zero) is explored. Zero Exponent and Roots

The logarithm is presented as an 18th century revelation. The natural exponential is compared to other exponential bases, graphically. Natural exponent

If the summation of a series grows faster than some rate, then it diverges. Up until the 17th century, no one knew whether the Harmonic Series diverged, or converged, as it grows very slowly, but faster than the Geometric series.

Tangents in general are considered, particularly those of curves which come to the minds of early modern natural philosophers. Tangents and their Calculus

The inverse operation of the tangent of a curve, in calculus, is the integral. Integral as Inverse of Tangent

In the modern frameworks, a ubiquitously appearing tool, is the representation of a complicated curve in terms of a polynomial with differential coefficients. Taylor Series