Geometric series with summation

3

1. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { N } r ( r + 1 ) ( 3 r + 4 ) = \frac { 1 } { 12 } N ( N + 1 ) ( N + 2 ) ( 9 N + 19 )$$
2. Express \(\frac { 3 r + 4 } { r ( r + 1 ) }\) in partial fractions and hence use the method of differences to find $$\sum _ { r = 1 } ^ { N } \frac { 3 r + 4 } { r ( r + 1 ) } \left( \frac { 1 } { 4 } \right) ^ { r + 1 }$$ in terms of \(N\).
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) } \left( \frac { 1 } { 4 } \right) ^ { r + 1 }\).

5.(a)Show that $$\sum _ { r = 0 } ^ { n } x ^ { - r } = \frac { x } { x - 1 } - \frac { x ^ { - n } } { x - 1 } \quad \text { where } x \neq 0 \text { and } x \neq 1$$ (b)Hence find an expression in terms of \(x\) and \(n\) for \(\sum _ { r = 0 } ^ { n } r x ^ { - ( r + 1 ) }\) for \(x \neq 0\) and \(x \neq 1\) Simplify your answer.
(c)Find \(\sum _ { r = 0 } ^ { n } \left( \frac { 3 + 5 r } { 2 ^ { r } } \right)\) Give your answer in the form \(a - \frac { b + c n } { 2 ^ { n } }\) ,where \(a , b\) and \(c\) are integers.

1. Show that

$$\sum _ { n = 2 } ^ { \infty } \left( \frac { 3 } { 4 } \right) ^ { n } \cos ( 180 n ) ^ { \circ } = \frac { 9 } { 28 }$$

7 Prove that \(\sum _ { r = 1 } ^ { n } \frac { r } { 2 ^ { r - 1 } } = 4 - \frac { n + 2 } { 2 ^ { n - 1 } }\) for all \(n \geqslant 1\).

A sequence \(\{u_n\}\) is given by $$u_1 = k$$ $$u_{2n} = u_{2n-1} \times p \qquad n \geq 1$$ $$u_{2n+1} = u_{2n} \times q \qquad n \geq 1$$ where \(k\), \(p\) and \(q\) are positive constants with \(pq \neq 1\)

1. Write down the first 6 terms of this sequence. [3]
2. Show that \(\sum_{r=1}^{2n} u_r = \frac{k(1+p)(1-(pq)^n)}{1-pq}\) [6]

In part (c) \([x]\) means the integer part of \(x\), so for example \([2.73] = 2\), \([4] = 4\) and \([0] = 0\)

1. Find \(\sum_{r=1}^{\infty} 6 \times \left(\frac{4}{3}\right)^{\left[\frac{r}{2}\right]} \times \left(\frac{3}{5}\right)^{\left[\frac{r-1}{2}\right]}\) [4]

[Total 13 marks]