OK.. I seemed to think that the series method gets the correct result. But
it does not entirely...
>>> import sympy as sy
>>> x = Symbol("x")
# Basic calling of series method is "sy.series(function, x0=..., n=...)"
where x0 is the point at which the expansion is taken at, while n is the
order of the *error*! (*not* the number of terms *prior* to the error
term!!) Other tools I have used imply that the error is of order (n+1) when
n is specified, so this is sort of confusing.
>>> sy.series(sy.sin(x),x0=0)
x - x**3/6 + x**5/120 + O(x**6)
>>> sy.series(sy.sin(x),x0=sy.pi/2)
1 - (x - pi/2)**2/2 + (x - pi/2)**4/24 + O((x - pi/2)**6, (x, pi/2))
>>> sy.series(sy.sin(x),x0=sy.pi)
pi + (x - pi)**3/6 - (x - pi)**5/120 - x + O((x - pi)**6, (x, pi))
# The above seems to be ok. But what if I expand to a different number of
terms than the default?
>>> sy.series(sy.sin(x),x0=sy.pi/2,n=1)
O(x - pi/2, (x, pi/2))
# The result *should* have been 1 + O(**1)! The value for sin(pi/2) is
missing.
>>> sy.series(sy.sin(x),x0=sy.pi/2,n=2)
1 + O((x - pi/2)**2, (x, pi/2))
# This result is ok.
>>> sy.series(sy.sin(x),x0=sy.pi/2,n=3)
1 - (x - pi/2)**2/2 + O((x - pi/2)**3, (x, pi/2))
# This result is ok
>>> sy.series(sy.sin(x),x0=sy.pi/2,n=4)
1 - (x - pi/2)**2/2 + O((x - pi/2)**4, (x, pi/2))
# This result is ok
>>> sy.series(sy.sin(x),x0=sy.pi/2,n=5)
1 - (x - pi/2)**2/2 + (x - pi/2)**4/24 + O((x - pi/2)**5, (x, pi/2))
# This result is ok
--
In *summary*:
* there is an error when n=0, i.e., no error: the result should have been
given as 1.
* it is also slightly confusing the way the number n of terms is defined --
this definition seems to be different from other tools...
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