Question 4 - A-Level Maths

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2019
Session	June
Topic	Sequences and series, recurrence and convergence

Use the method of differences to show that \(\sum _ { r = 1 } ^ { N } \frac { 1 } { ( 3 r + 1 ) ( 3 r - 2 ) } = \frac { 1 } { 3 } - \frac { 1 } { 3 ( 3 N + 1 ) }\).
Find the limit, as \(N \rightarrow \infty\), of \(\sum _ { r = N + 1 } ^ { N ^ { 2 } } \frac { N } { ( 3 r + 1 ) ( 3 r - 2 ) }\).