Surd rationalization method of differences

3

1. By simplifying \(\left( x ^ { n } - \sqrt { x ^ { 2 n } + 1 } \right) \left( x ^ { n } + \sqrt { x ^ { 2 n } + 1 } \right)\), show that \(\frac { 1 } { x ^ { n } - \sqrt { x ^ { 2 n } + 1 } } = - x ^ { n } - \sqrt { x ^ { 2 n } + 1 }\). [1]
   Let \(u _ { n } = x ^ { n + 1 } + \sqrt { x ^ { 2 n + 2 } + 1 } + \frac { 1 } { x ^ { n } - \sqrt { x ^ { 2 n } + 1 } }\).
2. Use the method of differences to find \(\sum _ { \mathrm { n } = 1 } ^ { \mathrm { N } } \mathrm { u } _ { \mathrm { n } }\) in terms of \(N\) and \(x\).
3. Deduce the set of values of \(x\) for which the infinite series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ is convergent and give the sum to infinity when this exists.

1. (a) Show that for \(r \geqslant 1\)

$$\frac { r } { \sqrt { r ( r + 1 ) } + \sqrt { r ( r - 1 ) } } \equiv A ( \sqrt { r ( r + 1 ) } - \sqrt { r ( r - 1 ) } )$$ where \(A\) is a constant to be determined.
(b) Hence use the method of differences to determine a simplified expression for $$\sum _ { r = 1 } ^ { n } \frac { r } { \sqrt { r ( r + 1 ) } + \sqrt { r ( r - 1 ) } }$$ (c) Determine, as a surd in simplest form, the constant \(k\) such that $$\sum _ { r = 1 } ^ { n } \frac { k r } { \sqrt { r ( r + 1 ) } + \sqrt { r ( r - 1 ) } } = \sqrt { \sum _ { r = 1 } ^ { n } r }$$

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

1. Show that, for \(r \geqslant 2\) $$\frac { 2 } { \sqrt { r } + \sqrt { r - 2 } } = \sqrt { r } - \sqrt { r - 2 }$$
2. Hence use the method of differences to determine $$\sum _ { r = 2 } ^ { n } \frac { 2 } { \sqrt { r } + \sqrt { r - 2 } }$$ giving your answer in simplest form.
3. Hence show that $$\sum _ { r = 4 } ^ { 50 } \frac { 2 } { \sqrt { r } + \sqrt { r - 2 } } = A + B \sqrt { 2 } + C \sqrt { 3 }$$ where \(A\), \(B\) and \(C\) are integers to be determined.

1 Given that $$u _ { k } = \frac { 1 } { \sqrt { } ( 2 k - 1 ) } - \frac { 1 } { \sqrt { } ( 2 k + 1 ) }$$ express \(\sum _ { k = 13 } ^ { n } u _ { k }\) in terms of \(n\). Deduce the value of \(\sum _ { k = 13 } ^ { \infty } u _ { k }\).