sol = 
        Solve[(2/η) (∑_(k = 0)^1)(((((-1))^k) (η^(-4 k (k + 1))))/(2 k + 1)) - (π/(2 (Sqrt[5]))) == 
            0, η];

pol = (First@First[η /. sol])[η]

Plot[pol, {η, 0, 3}]

(4/(π^2)) Log[η] /. 
        Select[Select[sol, 
            Im[(η /. #)] == 0 &], (η /. #) > 1 &] // 
      N