Summation of series

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

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

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

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

By tossing a coin one of three princes is chosen to be the next King of Randomia. Does each prince have an equal chance of taking the throne?

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

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

Can you find the sum of the squared sine values?

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?