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What is the reason behind calculating GCD in Pollard rho integer factorisation?

This is the pseudo code for calculating integer factorisation took from CLRS. But what is the point in calculating GCD involved in Line 8 and the need for doubling k when i == k in Line 13.? Help please....

sum of series AP GP clrs appendix A.1-4

I am trying to prove an equation given in the CLRS exercise book. The equation is: Sigma k=0 to k=infinity (k-1)/2^k = 0 I solved the LHS but my answer is 1 whereas the RHS should be 0 Following is my solution: Let's say S = k/2^k = 1/2 +...