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I would like to do some abstract power series manipulation. I am defining a power series like this

func = Sum[Subscript[a,i] * x^i, {i, 0, M}];

When I try to take the derivative I find

D[func, x] # Sum[i x^(-1 + i) Subscript[a, i], {i, 0, Infinity}]

Thus, the problem is that the index for sum still starts at zero as opposed to one. I am hoping some one can demonstrate a way to handle this issue, or point me toward a better way to handle abstract series.

This leads to problems when replacing with zero. Evaluating the expression below throws, "Power: Infinite expression 1/0 encountered" error.

D[func, x] /. x -> 0

Edit: An interesting thing I've found for higher derivatives is that mathematica will find ways to ensure that the proper terms are zero. In the following example, the 0th and 1st term of the second derivative of the power series is correctly zero.

D[Sum[Subscript[a, i] x^i, {i, 0, M}], {x, 2}] #Sum[i*Subscript[a, i]*(i*x^(i - 2) - x^(i - 2)), {i, 0, M}]
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    $\begingroup$ When dealing with sums, I often find it convenient to make the summation "implicit", dealing with the terms of the sum without writing the Sum itself. Clearly, you then need to keep track of the limits of the sum yourself. Without knowing what you plan to do next, it's difficult to make a definite suggestion. $\endgroup$ Commented Jul 14 at 21:49
  • $\begingroup$ @mikado I would like to consider derivatives of a fraction two multivariate power series. In other words, d/dx F(x,y) where F(x,y)=h(x,y)/g(x,y) and h and g are analytic at 0. I’m then interest in considering existence of the derivative at 0 using derivative exists iff lim h-> 0 ( f(h)-f(0) -Df(0)h ) / ||h|| = 0 $\endgroup$ Commented Jul 14 at 22:08
  • $\begingroup$ @Nasser that is what I get too. The problem is that the index should start at 1 not zero after taking the derivative. I just copied the wrong input/output pair $\endgroup$ Commented Jul 14 at 22:34
  • $\begingroup$ @mikado Besides looking for a more elegant way to handle power series, the crux of the issue wrt my current implementation can be seen in the additional information I have added. Specifically, D[ func, x] /. x->0 throws 1/0 error $\endgroup$ Commented Jul 14 at 22:54
  • $\begingroup$ @Nasser It is true that $\partial_x \sum_{n,m=0}^\infty a_{nm} \,x^n y^m = \sum_{n,m=0}^\infty n a_{nm} \,x^{n-1} y^m$ for all $x \neq 0$. The problem is when replacing x with zero this becomes undefined which is why it is normally written with index n starting from one not zero $\endgroup$ Commented Jul 15 at 1:58

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