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

[n=1, ∞]∑⌊cos (π/n) + 1⌋/2^n

That this series is convergent follows immediately from the fact that the ⌊cos (π/n) + 1⌋ is bounded and 1/2^n is absolutely convergent. To find the sum, first we investigate the behavior of ⌊cos (π/n) + 1⌋:

If n=1, then ⌊cos (π/n) + 1⌋ = ⌊cos (π) + 1⌋ = ⌊-1 + 1⌋ = ⌊0⌋ = 0
If n≥2, then 0 ≤ cos (π/n) < 1, so 1 ≤ cos (π/n) + 1 < 2, so ⌊cos (π/n) + 1⌋ = 1.

Thus the series can be rewritten as:

0 + [n=2, ∞]∑1/2^n

Using the formula for the sum of a geometric series, this is:

0 + (1/4)/(1 - 1/2)
(1/4)/(1/2)
1/2