Golden ratio problems

@Vijay Simha – I'll comment a proper write up here itself, you can copy-paste it there.

@Vijay Simha –

Original:

Suppose the sum of reciprocals of cubes of all the triangular numbers converges to a number A

1 + 1/3^3 + 1/6^3 + 1/10^3 + 1/15^3 + 1/21^3 + 1/28^3 + 1/36^3 + 1/45^3 + .... = A

and that the sum of reciprocals of squares of all the triangular numbers converges to B

1 + 1/3^2 + 1/6^2 + 1/10^2 + 1/15^2 + 1/21^2 + 1/28^2 + 1/36^2 + 1/45^2 + ....= B

and A/B can be expressed as M*(P - pi^2)/(pi^2 - O) where M, O and P are all integers.

What is the value of M + O + P ?

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Latexed and edited:

If the sum of reciprocals of cubes of all the triangular numbers converges to the number $A$ ,

$A = 1 + \dfrac{1}{3^3} + \dfrac{1}{6^3} + \dfrac{1}{{10}^3} + \dfrac{1}{{15}^3} + \dfrac{1}{{21}^3} + \dfrac{1}{{28}^3} + \dfrac{1}{{36}^3} + \dfrac{1}{{45}^3} + \cdots$

and the sum of reciprocals of squares of all the triangular numbers converges to the number $B$ ,

$B = 1 + \dfrac{1}{3^2} + \dfrac{1}{6^2} + \dfrac{1}{{10}^2} + \dfrac{1}{{15}^2} + \dfrac{1}{{21}^2} + \dfrac{1}{{28}^2} + \dfrac{1}{{36}^2} + \dfrac{1}{{45}^2} + \cdots$

then $\dfrac AB$ can be expressed as $M \dfrac{\left( P - \pi^2 \right)}{\left( \pi^2 - O \right)}$ where $M, O$ and $P$ are positive integers. Find $M+O+P$ .

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Raw format for the Latexes:

If the sum of reciprocals of cubes of all the triangular numbers converges to the number *(A*),

*[A = 1 + \dfrac{1}{3^3} + \dfrac{1}{6^3} + \dfrac{1}{{10}^3} + \dfrac{1}{{15}^3} + \dfrac{1}{{21}^3} + \dfrac{1}{{28}^3} + \dfrac{1}{{36}^3} + \dfrac{1}{{45}^3} + \cdots *]

and the sum of reciprocals of squares of all the triangular numbers converges to the number *(B*),

*[B = 1 + \dfrac{1}{3^2} + \dfrac{1}{6^2} + \dfrac{1}{{10}^2} + \dfrac{1}{{15}^2} + \dfrac{1}{{21}^2} + \dfrac{1}{{28}^2} + \dfrac{1}{{36}^2} + \dfrac{1}{{45}^2} + \cdots *]

then *(\dfrac AB*) can be expressed as *(M \dfrac{\left( P - \pi^2 \right)}{\left( \pi^2 - O \right)}*) where *(M, O*) and *(P*) are positive integers. Find *(M+O+P*).

Note: Before copy-pasting the above Latex formatted write-up remove all the asterix symbols (*) and replace them with a backslash ( \ ).