Infinite series

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

Is it true that a large integer m can be taken such that: 1 + 1/2 + 1/3 + ... +1/m > 100 ?

How much of the square is coloured blue? How will the pattern continue?

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.