Question 8 - A-Level Maths

Exam Board	OCR
Module	FP1 (Further Pure Mathematics 1)
Year	2007
Session	January
Topic	Sequences and series, recurrence and convergence

Show that $( r + 2 ) ! - ( r + 1 ) ! = ( r + 1 ) ^ { 2 } \times r !$.
Hence find an expression, in terms of $n$, for $$2 ^ { 2 } \times 1 ! + 3 ^ { 2 } \times 2 ! + 4 ^ { 2 } \times 3 ! + \ldots + ( n + 1 ) ^ { 2 } \times n ! .$$
State, giving a brief reason, whether the series $$2 ^ { 2 } \times 1 ! + 3 ^ { 2 } \times 2 ! + 4 ^ { 2 } \times 3 ! + \ldots$$ converges.