Given series: cot1 3 cot1 7 cot1 13 cot1 21 ...

To find: Sum of first 50 terms of the series

Solution:

We can write the given series as:

cot1 3 cot1 7 cot1 13 cot1 21 ... = tan1 2 (cot1 3 - cot1 2) + tan1 2 (cot1 7 - cot1 6) + tan1 2 (cot1 13 - cot1 12) + tan1 2 (cot1 21 - cot1 20) + ...

Now, we need to find the sum of first 50 terms of the series. Thus, we need to find the sum of the following 50 terms:

tan1 2 (cot1 3 - cot1 2), tan1 2 (cot1 7 - cot1 6), tan1 2 (cot1 13 - cot1 12), tan1 2 (cot1 21 - cot1 20), ...

We can simplify each term using the formula:

cot(A) - cot(B) = tan(B - A)

Using this formula, we get:

tan1 2 (cot1 3 - cot1 2) = tan1 2 (tan1 3 - tan1 2) = tan1 2 (tan1 3 - tan1 2)

Similarly, we can simplify the other terms as:

tan1 2 (cot1 7 - cot1 6) = tan1 2 (tan1 6 - tan1 7) = -tan1 2 (tan1 7 - tan1 6)

tan1 2 (cot1 13 - cot1 12) = tan1 2 (tan1 12 - tan1 13) = -tan1 2 (tan1 13 - tan1 12)

tan1 2 (cot1 21 - cot1 20) = tan1 2 (tan1 20 - tan1 21) = -tan1 2 (tan1 21 - tan1 20)

We can observe that the terms in the series occur in pairs of opposite signs. Thus, we can pair the terms as follows:

{tan1 2 (tan1 3 - tan1 2) - tan1 2 (tan1 6 - tan1 7)} + {tan1 2 (tan1 9 - tan1 8) - tan1 2 (tan1 12 - tan1 13)} + {tan1 2 (tan1 15 - tan1 14) - tan1 2 (tan1 18 - tan1 19)} + ...

Each pair of terms can be simplified as:

tan(A) - tan(B) = (tan(A) - tan(B))/(1 + tan(A)tan(B))

Using this formula, we get:

{tan1 2 (tan1 3 - tan1 2) - tan1 2 (tan1 6 - tan1 7)} = (tan1 3 - tan1 7)/(1 + tan1 3 tan1 7)

Similarly, we can simplify the other pairs of terms as:

{tan1 2 (tan1 9