Given the modified Bessel functions of the first kind $I_n(\cdot)$, it appears that $$ \lim_{x \rightarrow \infty} \left[ x \left( \frac{I_1(x^2)}{I_0(x^2)} - \sum_{n=1}^\infty \frac{I_{2n+1}(x^2)}{n(n+1)I_0(x^2)} \right) \right] = K \approx 2.527... $$ However I was unable to identify the constant $K$ by the usual Bessel function identities.
Does anyone with more experience than me know of any tricks to get this number out?