Question 2 - A-Level Maths

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2018
Session	June
Topic	Sequences and series, recurrence and convergence

Verify that $$\frac { n ( \mathrm { e } - 1 ) + \mathrm { e } } { n ( n + 1 ) \mathrm { e } ^ { n + 1 } } = \frac { 1 } { n \mathrm { e } ^ { n } } - \frac { 1 } { ( n + 1 ) \mathrm { e } ^ { n + 1 } }$$ Let $S _ { N } = \sum _ { n = 1 } ^ { N } \frac { n ( \mathrm { e } - 1 ) + \mathrm { e } } { n ( n + 1 ) \mathrm { e } ^ { n + 1 } }$.
Express $S _ { N }$ in terms of $N$ and e.
Let $S = \lim _ { N \rightarrow \infty } S _ { N }$.
Find the least value of $N$ such that $( N + 1 ) \left( S - S _ { N } \right) < 10 ^ { - 3 }$.