If the sum of the series ( 1 2 - 1 3 ) + ( 1 2^2 - 1 2 3 + 1 3^2 ) + ( 1 2^3 - 1 2^2…

MATHEMATICS · Subjective · A Level · Mathematics · Sequences and series · Sequences and series

Problem

If the sum of the series (12−13)+(122−12⋅3+132)+(123−122⋅3+12⋅32−133)\left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{2^2} - \frac{1}{2 \cdot 3} + \frac{1}{3^2}\right) + \left(\frac{1}{2^3} - \frac{1}{2^2 \cdot 3} + \frac{1}{2 \cdot 3^2} - \frac{1}{3^3}\right) +(124−123⋅3+122⋅32−12⋅33+134)+…+ \left(\frac{1}{2^4} - \frac{1}{2^3 \cdot 3} + \frac{1}{2^2 \cdot 3^2} - \frac{1}{2 \cdot 3^3} + \frac{1}{3^4}\right) + \dots is αβ\frac{\alpha}{\beta}, where α\alpha and β\beta are co-prime, then α+3β\alpha + 3\beta is equal to -

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