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 · 3 + 1 2 · 3 2 - 1 3 3 + 1 2 4 - 1 2 3 · 3 + 1 2 2 · 3 2 - 1 2 · 3 3 + 1 3 4 + . . . . . . is α β , where α and β are co-prime, then α + 3 β is equal to ________.

Step-by-step solution

Given, Series 1 2 - 1 3 + 1 2 2 - 1 2 · 3 + 1 3 2 + 1 2 3 - 1 2 2 · 3 + 1 2 · 3 2 - 1 3 3 + 1 2 4 - 1 2 3 · 3 + 1 2 2 · 3 2 - 1 2 · 3 3 + 1 3 4 + . . . . . . Now let a = 1 2 ,   b = 1 3 So, series will be: a - b + a 2 - a b + b 2 + a 3 - a 2 b + a b 2 - b 3 + . . . . . = 1 a + b a 2 - b 2 + a 3 + b 3 + a 4 - b 4 + … = 1 a + b a 2 + a 3 + a 4 … - b 2 - b 3 + b 4 … . Now using infinite G . P formula we get, = 1 a + b a 2 + a 3 + a 4 … - b 2 - b 3 + b 4 &#8230