Limits

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Find all the turning points of y=x^{1/x} for x>0 and decide whether each is a maximum or minimum. Give a sketch of the graph.

Prove that the sum of the reciprocals of the first n triangular numbers gets closer and closer to 2 as n grows.

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

Find the maximum value of n to the power 1/n and prove that it is a maximum.

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

A finite area inside and infinite skin! You can paint the interior of this fractal with a small tin of paint but you could never get enough paint to paint the edge.

Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers.