7 The quadratic equation \(x ^ { 2 } + 2 k x + k = 0\), where \(k\) is a non-zero constant, has roots \(\alpha\) and \(\beta\). Find a quadratic equation with roots \(\frac { \alpha + \beta } { \alpha }\) and \(\frac { \alpha + \beta } { \beta }\).
- Show that \(\frac { 1 } { \sqrt { r + 2 } + \sqrt { r } } \equiv \frac { \sqrt { r + 2 } - \sqrt { r } } { 2 }\).
- Hence find an expression, in terms of \(n\), for
$$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r + 2 } + \sqrt { r } }$$
- State, giving a brief reason, whether the series \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { \sqrt { r + 2 } + \sqrt { r } }\) converges.