2.1.2 Power series

Taylor series:

sage: var('f0 k x')
(f0, k, x)
sage: g = f0/sinh(k*x)^4
sage: g.taylor(x, 0, 3)
f0/(k^4*x^4) - 2*f0/(3*k^2*x^2) + 11*f0/45 - 62*k^2*f0*x^2/945
sage: maxima(g).powerseries('x',0)
16*f0*('sum((2^(2*i1-1)-1)*bern(2*i1)*k^(2*i1-1)*x^(2*i1-1)/(2*i1)!,i1,0,inf))^4
Of course, you can view the LaTeX-ed version of this using view(g.powerseries('x',0)).

The Maclaurin and power series of $ \log({\frac{\sin(x)}{x}})$ :

sage: f = log(sin(x)/x)
sage: f.taylor(x, 0, 10)
-x^2/6 - x^4/180 - x^6/2835 - x^8/37800 - x^10/467775
sage: [bernoulli(2*i) for i in range(1,7)]
[1/6, -1/30, 1/42, -1/30, 5/66, -691/2730]
sage: maxima(f).powerseries(x,0)
('sum((-1)^i2*2^(2*i2)*bern(2*i2)*x^(2*i2)/(i2*(2*i2)!),i2,1,inf))/2

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