determine whether the series is convergent or divergent. if the series is convergent, find its sum.?

Question

please help me do this problem,please,,,

summation notation symbol(n=1 to infinity) of { greatest integer function symbol, (cos pi/n + 1)}/2^n

Answer 1

Let [x] denote the greatest integer less or equal x (the floor of x) and let a_n be the general term of the series.Then,

n = 1 => a_1 = [cos(pi + 1]/2 = [-1 +1]/2 = [0]/2 = 0/2 = 0

If n>=2, then 0 < pi/n <= pi/2, so that 0 <= cos(pi/n) < 1
Therefore, 1 <= 1 + cos(pi/n) < 2 and [1 + cos(pin)] = 1. It then follows that a_n = 1/2^n.

Our sum is S = a_1 + a_2 +...a_n.....= 0 + 1/2....+1/2^n.... = 1/2 ...+1/2^n...= Sum(n=1, oo) 1/2^n, so a geometric series that converges because it common ratio is 1/2 and |1/2| <1. Hence, S = (1/2)/(1 - 1/2) = 1.