Standard +0.3 This is a straightforward telescoping series question where the exponential terms cancel systematically. The first part requires recognizing the telescoping pattern (standard FP1 technique), and the second part involves basic convergence analysis using limits of exponential functions. While it's a Further Maths topic, the execution is mechanical with no novel insight required.
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.
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.
\hfill \mbox{\textit{CAIE FP1 2002 Q1 [5]}}