$\displaystyle \text{Ci} (x) = -\int_{x}^{\infty} \frac{\cos t}{t} \ dt $

$\displaystyle = -\int_{x}^{\infty} \frac{1}{t} \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n!)} \ t^{2n} \ dt $

$\displaystyle = -\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n)!} \int_{x}^{\infty} t^{2n-1} \ dt $ (since the series expansion of the cosine function converges uniformly)

$\displaystyle = -\int^{\infty}_{x} \frac{dt}{t} -\sum_{n=1}^{\infty} \frac{(-1)^{n}}{(2n)!} \int_{x}^{\infty} t^{2n-1} \ dt $

$\displaystyle = -\ln t\Big|^{\infty}_{x} - \frac{1}{2} \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n (2n)!} \ t^{2n} \Big|_{x}^{\infty} $

$\displaystyle \lim_{t \to \infty} \Big(-\ln t - \frac{1}{2} \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n (2n)!} t^{2n} \Big) + \ln x + \frac{1}{2} \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n (2n)!} \ x^{2n} $

I thought this had to be incorrect, but Maple says that the limit does indeed evaluate to Euler's Constant. Why?