Proofs for series forms of $\operatorname{Si}(x)$, $\operatorname{Ci}(x)$ and $\operatorname{Li}(x)$ I am studying the book Gamma by Julian Havil and there are three equations below stated without proofs: 
\begin{align}
\operatorname {Si}(x) & = \sum_{k = 1}^\infty (-1)^{k - 1}  \frac{x^{2k - 1}}{(2k-1)(2k-1)!}, \\[10pt]
\operatorname{Ci}(x) & = -\gamma - \ln x -\sum_{k=1}^\infty \frac{(-x^2)^k}{(2k)(2k)!}, \\[10pt]
\operatorname{Li}(x) & = \gamma + \ln \ln x + \sum_{k=1}^\infty \frac{\ln^r x}{kk!}.
\end{align}
I couldn't find any proof for the three equations above. 
A clear proof for each either in MSE or a book containing them would be much appreciated.
 A: The only book I know of that directly derives these formulae is Theorie Des Integrallogarithmus Und Verwandter Transzendenten by Niels Nielsen.  This is a German language book published in 1906.  It is freely available online if I remember correctly.  
But also note that one can alternatively define
$$\operatorname{Si}(x) = \int_0^x \frac{\sin{t}}{t} \, \text{d}t$$
so that term-by-term integration of the Taylor series for $\sin$ yields
$$\operatorname{Si}(x) = \sum_{n=1}^\infty \frac{(-1)^{n-1} x^{2n-1}}{(2n-1)(2n-1)!}$$
The trickier part is to show that
$$\operatorname{Ci}(x) = -\int_x^\infty \frac{\cos t}{t} \, \text{d}t = \gamma + \log x + \int_0^x \frac{\cos t-1}{t} \, \text{d}t$$
so that one can integrate term-by-term to arrive at
$$\operatorname{Ci}(x) = \gamma + \log x + \sum_{n=1}^\infty \frac{(-1)^{n-1} x^{2n}}{2n(2n)!}$$
But this is all done in Nielsen using $\operatorname{Li}$.
Edit: I realize this book is not great for self-study, but it is truly a gem and if you are interested in this sort of mathematics it is a great resource.
