Standard +0.8 This is a Further Maths question combining partial fractions with method of differences applied to a geometric series. While partial fractions is routine, recognizing how to telescope the series with the geometric factor (1/3)^(n+1) requires insight beyond standard techniques. The final limit evaluation adds another layer, making this moderately challenging but within reach for FP1 students.
2 Express
$$\frac { 2 n + 3 } { n ( n + 1 ) }$$
in partial fractions and hence use the method of differences to find
$$\sum _ { n = 1 } ^ { N } \frac { 2 n + 3 } { n ( n + 1 ) } \left( \frac { 1 } { 3 } \right) ^ { n + 1 }$$
in terms of \(N\).
Deduce the value of
$$\sum _ { n = 1 } ^ { \infty } \frac { 2 n + 3 } { n ( n + 1 ) } \left( \frac { 1 } { 3 } \right) ^ { n + 1 }$$
2 Express
$$\frac { 2 n + 3 } { n ( n + 1 ) }$$
in partial fractions and hence use the method of differences to find
$$\sum _ { n = 1 } ^ { N } \frac { 2 n + 3 } { n ( n + 1 ) } \left( \frac { 1 } { 3 } \right) ^ { n + 1 }$$
in terms of $N$.
Deduce the value of
$$\sum _ { n = 1 } ^ { \infty } \frac { 2 n + 3 } { n ( n + 1 ) } \left( \frac { 1 } { 3 } \right) ^ { n + 1 }$$
\hfill \mbox{\textit{CAIE FP1 2007 Q2 [5]}}