  In this paper we give an elementary proof of the alternative formulation of
Ramanujan's formula for $ \zeta(2m +1)$ in terms of hyperbolic cotangent sums:
$$\sum_{n = 1}^{\infty}\dfrac{\coth(\pi n)}{n^{4m + 3}} = 2\,(2\pi)^{4m +
3}\sum_{k = 0}^{2m + 2}(-1)^{k+1}\,\dfrac{\mathcal{B}_{2k}\,\mathcal{B}_{4m + 4
-2k}}{4\,(2k)!\,(4m+4-2k)!}$$ using partial fraction expansion of cotangent and
Euler's formula for $ \zeta(2m) $.
