Three linear factors in denominator

2. (a) Write \(\frac { 3 r + 1 } { r ( r - 1 ) ( r + 1 ) }\) in partial fractions.
(b) Hence find $$\sum _ { r = 2 } ^ { n } \frac { 3 r + 1 } { r ( r - 1 ) ( r + 1 ) } \quad n \geqslant 2$$ giving your answer in the form $$\frac { a n ^ { 2 } + b n + c } { 2 n ( n + 1 ) }$$ where \(a\), \(b\) and \(c\) are integers to be determined.
(c) Hence determine the exact value of $$\sum _ { r = 15 } ^ { 20 } \frac { 3 r + 1 } { r ( r - 1 ) ( r + 1 ) }$$

5. (a) Express \(\frac { 2 } { r ( r + 1 ) ( r + 2 ) }\) in partial fractions.
(b) Using your answer to part (a) and the method of differences, show that $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { n ( n + 3 ) } { 2 ( n + 1 ) ( n + 2 ) }$$

1. (a) Express \(\frac { 4 r + 2 } { r ( r + 1 ) ( r + 2 ) }\) in partial fractions.
   (b) Hence, using the method of differences, prove that

$$\sum _ { r = 1 } ^ { n } \frac { 4 r + 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { n ( a n + b ) } { 2 ( n + 1 ) ( n + 2 ) }$$ where \(a\) and \(b\) are constants to be found.

8

1. Show that \(\frac { 1 } { r } - \frac { 3 } { r + 1 } + \frac { 2 } { r + 2 } \equiv \frac { 2 - r } { r ( r + 1 ) ( r + 2 ) }\).
2. Hence show that \(\sum _ { r = 1 } ^ { n } \frac { 2 - r } { r ( r + 1 ) ( r + 2 ) } = \frac { n } { ( n + 1 ) ( n + 2 ) }\).
3. Find the value of \(\sum _ { r = 2 } ^ { \infty } \frac { 2 - r } { r ( r + 1 ) ( r + 2 ) }\).

2 Express \(\frac { 4 } { r ( r + 1 ) ( r + 2 ) }\) in partial fractions and hence find \(\sum _ { r = 1 } ^ { n } \frac { 4 } { r ( r + 1 ) ( r + 2 ) }\). Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 4 } { r ( r + 1 ) ( r + 2 ) }\).

1 Express \(\frac { 1 } { r ( r + 1 ) ( r - 1 ) }\) in partial fractions. Find $$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$ State the value of $$\sum _ { r = 2 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$

1. The function \(f\) is given by

$$f ( x ) = \frac { 25 x + 32 } { ( 2 x - 5 ) ( x + 1 ) ( x + 2 ) }$$

1. Express \(f ( x )\) in terms of partial fractions.
2. Show that \(\int _ { 1 } ^ { 2 } f ( x ) \mathrm { d } x = - \ln P\), where \(P\) is an integer whose value is to be found. \section*{
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3. Show that the sign of \(f ( x )\) changes in the interval \(x = 2\) to \(x = 3\). Explain why the change of sign method fails to locate a root of the equation \(f ( x ) = 0\) in this case.