CAIE
FP1
2007
November
Q2
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
$$\sum _ { n = 1 } ^ { \infty } \frac { 2 n + 3 } { n ( n + 1 ) } \left( \frac { 1 } { 3 } \right) ^ { n + 1 }$$