Question 1 - A-Level Maths

Exam Board	OCR
Module	Further Pure Core 2 (Further Pure Core 2)
Year	2024
Session	June
Topic	Sequences and series, recurrence and convergence

Use the method of differences to show that \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \left( \frac { 1 } { \mathrm { r } } - \frac { 1 } { \mathrm { r } + 1 } \right) = 1 - \frac { 1 } { \mathrm { n } + 1 }\).
Hence determine the following sums.
1. \(\quad \sum _ { r = 1 } ^ { 99 } \frac { 1 } { r } - \frac { 1 } { r + 1 }\)
2. \(\quad \sum _ { r = 100 } ^ { \infty } \frac { 1 } { r } - \frac { 1 } { r + 1 }\)