Method of differences with exponential terms

Questions using the method of differences to sum series involving exponential functions e^(rx) or powers of constants like 2^r, including determining convergence conditions for infinite series.

2 questions · Standard +0.8

4.06b Method of differences: telescoping series
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CAIE Further Paper 1 2022 June Q4
9 marks Challenging +1.2
4 Let \(\mathrm { u } _ { \mathrm { r } } = \mathrm { e } ^ { \mathrm { rx } } \left( \mathrm { e } ^ { 2 \mathrm { x } } - 2 \mathrm { e } ^ { \mathrm { x } } + 1 \right)\).
  1. Using the method of differences, or otherwise, find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \mathrm { u } _ { \mathrm { r } }\) in terms of \(n\) and \(x\).
  2. Deduce the set of non-zero values of \(x\) for which the infinite series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ is convergent and give the sum to infinity when this exists.
  3. Using a standard result from the list of formulae (MF19), find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \ln \mathrm { u } _ { \mathrm { r } }\) in terms of \(n\) and \(x\).
CAIE FP1 2002 November Q1
5 marks Standard +0.3
1 Given that $$u _ { n } = \mathrm { e } ^ { n x } - \mathrm { e } ^ { ( n + 1 ) x }$$ find \(\sum _ { n = 1 } ^ { N } \| _ { n }\) in terms of \(N\) and \(x\). Hence determine the set of values of \(x\) for which the infinite series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ is convergent and give the sum to infinity for cases where this exists.