Asymptotic expansion of cosine integral Can anybody help with this problem to find the full asymptotic expansion of 
$\int_1^\infty \frac{\cos(xt)\, d t}{t}$ from Bender & Orzsag). Does it work by Taylor  expansion?

Considering $$\int\frac{\cos (xt)}{t}\,dt=\text{Ci}(t x)$$ $$I=\int_1^\infty \frac{\cos (xt)}{t}\,dt=-\text{Ci}( x)$$ provided $x>0$.
Now, looking here, you will find that $$I=\cos(x)\left(\frac{1}{x^2}-\frac{6}{x^4}+\frac{120}{x^6}+O\left(\frac{1}{x^7}
   \right) \right)-\sin(x)\left(\frac{1}{x}-\frac{2}{x^3}+\frac{24}{x^5}+O\left(\frac{1}{x^7}\right) \right)$$
For illustration purposes, let us use it for $x=100 \pi$. The above expansion would give $$\frac{3-1500 \pi ^2+2500000 \pi ^4}{25000000000 \pi ^6}\approx 0.0000101315025301179$$ while the exact value should be $$0.0000101315025300648$$
