The value of the ratio  (1 + 122 + 132 + 142 + ….)/(1 − 122 + 132 − 142 + ….)  is  …… .........

The given series are geometric series.

The first series is 1 + 12^2 + 13^2 + 14^2 + ..., where the common ratio is 12^2.

The sum of an infinite geometric series is given by S = a / (1 - r), where a is the first term and r is the common ratio.

So, the sum of the first series is S1 = 1 / (1 - 12^2).

The second series is 1 - 12^2 + 13^2 - 14^2 + ..., where the common ratio is -12^2.

So, the sum of the second series is S2 = 1 / (1 - (-12^2)).

The value of the ratio (1 + 12^2 + 13^2 + 14^2 + ...)/(1 - 12^2 + 13^2 - 14^2 + ...) is therefore S1 / S2.

However, there seems to be a mistake in the question. The common ratio in a geometric series should be the same for all terms, but here it is increasing by 1 for each term. Therefore, these are not geometric series and the formula for the sum of an infinite geometric series cannot be applied.

Please check the question and provide the correct series.