Question about sum and diff

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When you write sum(diff(sin(x),x,n),n,1,10), you try to do a symbolic sum, hence the Python name n should be defined as a symbolic variable (an element of the Symbolic Ring). However, in Sage, the name n corresponds to the function that makes numerical_approx:

sage: n
<function numerical_approx at 0xb3f63684>
sage: n(pi)
3.14159265358979

This explains why you got the error TypeError: no canonical coercion from <type 'function'> to Symbolic Ring, this is because Sage try (without success) to transform the numerical_approx function into an element of the Symbolic Ring.

So you may try to overwrite the name n to correspond to the symbolic variable "n":

sage: n = SR.var('n')
sage: sum(diff(sin(x),x,n),n,1,10)
0

But then the result is unexpected ! The problem is that now, when you write diff(sin(x),x,n), Sage does not understands "differentiate n times relative to x", but "differentiate relative to x and then to n", so you get 0 since a function of x has a zero derivative relative to n.

For Sage to understand "differentiate n times relative to x", n needs to be an integer, not a symbol. So, instead you can do a non-symbolic sum of a list that contains all derivatives:

sage: f(x) = sum([diff(sin(x),x,n) for n in range(1,11)])
sage: f(x)
cos(x) - sin(x)

Which seems correct.