\(\cfrac{\sqrt[3]{n_2}-\sqrt[3]{n_1}}{\sqrt[3]{n_1n_2}}\ne\cfrac{1}{n_1^2}-\cfrac{1}{n_2^2}=\cfrac{n_2^2-n_1^2}{(n_1n_2)^2}\)
but are they parallel over the range of \(n_1\) and \(n_2\) in consideration.
A plot of \(\cfrac { \sqrt [ 3 ]{ n_{ 2 } } -\sqrt [ 3 ]{ n_{ 1 } } }{ \sqrt [ 3 ]{ n_{ 1 }n_{ 2 } } } -\cfrac { n_{ 2 }^{ 2 }-n_{ 1 }^{ 2 } }{ (n_{ 1 }n_{ 2 })^{ 2 } } \) gives
where the values of the expression for \(n_1=1\) is negative and the graphs of for \(n_1=2\) and \(n_1=5\) are coincidental.
A plot of the ratio \(\cfrac { \cfrac { \sqrt [ 3 ]{ n_{ 2 } } -\sqrt [ 3 ]{ n_{ 1 } } }{ \sqrt [ 3 ]{ n_{ 1 }n_{ 2 } } } }{ \cfrac { n_{ 2 }^{ 2 }-n_{ 1 }^{ 2 } }{ (n_{ 1 }n_{ 2 })^{ 2 } } } \) gives,
The values of the ratio varies with both \(n_2\) and \(n_1\). In the case of \(n_1=1\), the ratio \(\cfrac { \cfrac { \sqrt [ 3 ]{ n_{ 2 } } -\sqrt [ 3 ]{ n_{ 1 } } }{ \sqrt [ 3 ]{ n_{ 1 }n_{ 2 } } } }{ \cfrac { n_{ 2 }^{ 2 }-n_{ 1 }^{ 2 } }{ (n_{ 1 }n_{ 2 })^{ 2 } } } \) is close to \(1\)
Rydberg constant is not dead. Yet...
