Sigma Notation Manipulation

3

1. A curve has intrinsic equation \(s = 2 \ln \left( \frac { \pi } { \pi - 3 \psi } \right)\) for \(0 \leqslant \psi < \frac { 1 } { 3 } \pi\), where \(s\) is the arc length measured from a fixed point P and \(\tan \psi = \frac { \mathrm { d } y } { \mathrm {~d} x } . \mathrm { P }\) is in the third quadrant. The curve passes through the origin O , at which point \(\psi = \frac { 1 } { 6 } \pi . \mathrm { Q }\) is the point on the curve at which \(\psi = \frac { 3 } { 10 } \pi\).
   1. Express \(\psi\) in terms of \(s\), and sketch the curve, indicating the points \(\mathrm { O } , \mathrm { P }\) and Q .
   2. Find the arc length OQ .
   3. Find the radius of curvature at the point O .
   4. Find the coordinates of the centre of curvature corresponding to the point O .
2. 1. Find the surface area of revolution formed when the curve \(y = \frac { 1 } { 3 } \sqrt { x } ( x - 3 )\) for \(1 \leqslant x \leqslant 4\) is rotated through \(2 \pi\) radians about the \(y\)-axis.
   2. The curve in part (b)(i) is one member of the family \(y = \frac { 1 } { 9 } \lambda \sqrt { x } ( x - \lambda )\), where \(\lambda\) is a positive parameter. Find the equation of the envelope of this family of curves.

1．（a）By considering the series $$1 + t + t ^ { 2 } + t ^ { 3 } + \ldots + t ^ { n }$$ or otherwise，sum the series $$1 + 2 t + 3 t ^ { 2 } + 4 t ^ { 3 } + \ldots + n t ^ { n - 1 }$$ for \(t \neq 1\) ．
（b）Hence find and simplify an expression for $$1 + 2 \times 3 + 3 \times 3 ^ { 2 } + 4 \times 3 ^ { 3 } + \ldots + 2001 \times 3 ^ { 2000 }$$ （c）Write down an expression for both the sums of the series in part（a）for the case where \(t = 1\) ．

3．A sequence \(\left\{ u _ { n } \right\}\) is given by $$\begin{aligned} u _ { 1 } & = k & & \\ u _ { 2 n } & = u _ { 2 n - 1 } \times p & & n \geqslant 1 \\ u _ { 2 n + 1 } & = u _ { 2 n } \times q & & n \geqslant 1 \end{aligned}$$ where \(k , p\) and \(q\) are positive constants with \(p q \neq 1\)
（a）Write down the first 6 terms of this sequence．
（b）Show that \(\sum _ { r = 1 } ^ { 2 n } u _ { r } = \frac { k ( 1 + p ) \left( 1 - ( p q ) ^ { n } \right) } { 1 - p q }\) In part（c） \([ x ]\) means the integer part of \(x\) ，so for example \([ 2.73 ] = 2 , [ 4 ] = 4\) and \([ 0 ] = 0\)
（c）Find \(\sum _ { r = 1 } ^ { \infty } 6 \times \left( \frac { 4 } { 3 } \right) ^ { \left[ \frac { r } { 2 } \right] } \times \left( \frac { 3 } { 5 } \right) ^ { \left[ \frac { r - 1 } { 2 } \right] }\)

1 It is given that \(\sum _ { r = 1 } ^ { n } u _ { r } = n ^ { 2 } ( 2 n + 3 )\), where \(n\) is a positive integer.

1. Find \(\sum _ { r = n + 1 } ^ { 2 n } u _ { r }\).
2. Find \(u _ { r }\).

2 For \(x , y \in \mathbb { R }\), the function f is given by \(\mathrm { f } ( x , y ) = 2 x ^ { 2 } \mathrm { y } ^ { 7 } + 3 x ^ { 5 } y ^ { 4 } - 5 x ^ { 8 } y\).

1. Prove that \(\mathrm { xf } _ { \mathrm { x } } + \mathrm { yf } _ { \mathrm { y } } = \mathrm { nf }\), where \(n\) is a positive integer to be determined.
2. Show that \(\mathrm { xf } _ { \mathrm { xx } } + \mathrm { yf } _ { \mathrm { xy } } = ( \mathrm { n } - 1 ) \mathrm { f } _ { \mathrm { x } }\).

1 You are given that $$\begin{aligned} u _ { 1 } & = 1 \\ u _ { n + 1 } & = \frac { u _ { n } } { 1 + u _ { n } } \end{aligned}$$ Find the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\). Give your answers as fractions.

2

1. Evaluate \(\sum _ { r = 2 } ^ { 5 } \frac { 1 } { r - 1 }\).
2. Express the series \(2 \times 3 + 3 \times 4 + 4 \times 5 + 5 \times 6 + 6 \times 7\) in the form \(\sum _ { r = 2 } ^ { a } \mathrm { f } ( r )\) where \(\mathrm { f } ( r )\) and \(a\) are to be determined.