Summation of series

Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!

What is the sum of: 6 + 66 + 666 + 6666 ............+ 666666666...6 where there are n sixes in the last term?

Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.

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

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

Find the smallest value for which a particular sequence is greater than a googol.

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

How can visual patterns be used to prove sums of series?