The function powsqrt(p) returns the formal power series that is the square root of p.

The power series of p must have a nonzero first term ($p\left(0\right)\ne 0$) for its logarithm to be well-defined.

The command with(powseries,powsqrt) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{powseries}\right)\:$

$s≔\mathrm{powsqrt}\left(x^2+x+1\right)\:$

$\mathrm{tpsform}\left(s\,x\,7\right)$

$1+\frac{1}{2}x+\frac{3}{8}{x}^2-\frac{3}{16}{x}^3+\frac{3}{128}{x}^4+\frac{15}{256}{x}^5-\frac{57}{1024}{x}^6+O\left(x^7\right)$

$t≔\mathrm{powsqrt}\left(\exp \left(x\right)\right)\:$

$\mathrm{tpsform}\left(t\,x\,5\right)$

$1+\frac{1}{2}x+\frac{1}{8}{x}^2+\frac{1}{48}{x}^3+\frac{1}{384}{x}^4+O\left(x^5\right)$

$\mathrm{tpsform}\left(\mathrm{powsqrt}\left(\mathrm{powdiff}\left(\mathrm{powlog}\left(1+x\right)\right)\right)\,x\,5\right)$

$1-\frac{1}{2}x+\frac{3}{8}{x}^2-\frac{5}{16}{x}^3+\frac{35}{128}{x}^4+O\left(x^5\right)$