Adding with cousins

Number Theory Level pending

For all \(n\ge 2\), there are \({ p }_{ n }\) the prime number less than or equal to \(n\) and \({ q }_{ n }\) the next prime number larger than \(n\). For example: \(n=3\), \({ p }_{ 3 }=3\) and \({ q }_{ 3 }=5\).

Find the value of

\[\frac { 1 }{ { p }_{ 2 }{ q }_{ 2 } } +\frac { 1 }{ { p }_{ 3 }{ q }_{ 3 } } +\frac { 1 }{ { p }_{ 4 }{ q }_{ 4 } } +\frac { 1 }{ { p }_{ 5 }{ q }_{ 5 } } +\frac { 1 }{ { p }_{ 6 }{ q }_{ 6 } } +\frac { 1 }{ { p }_{ 7 }{ q }_{ 7 } } +\frac { 1 }{ { p }_{ 8 }{ q }_{ 8 } } +\frac { 1 }{ { p }_{ 9 }{ q }_{ 9 } } +\frac { 1 }{ { p }_{ 10 }{ q }_{ 10 } } \]

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