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Wolfram Mathematica 7 routinely confuses sums with integrals.This

Example 1:

DSolve[(-Log[Log[a]] f'[x] + f''[x])/(Log[a] f'[x]) == D[Sum[f[x], x], x], f[x], x]

g[x_] := f[x] /. s
g[x]


Checking the result by inserting it into the equation shows the result is just one exampleincorrect:

http://www.wolframalpha.com/input/?i=derivative+sum+2^(1/k)+from+0+to+x-1

$$\frac{\partial}{\partial x}\sum_0^{x-1} 2^{1/k} (-Log[Log[a]] g'[x] + g''[x])/(Log[a] g'[x]) - D[Sum[g[x], x], x]  Example 2: s=NDSolve[{0.9159460564995328*Derivative[1][f][x] =2^{1/k}$$= f[x]*Product[f[x], x], f[0] == 1}, f, {x, -1.9, 15}]

Plot[Evaluate[f[x] /. s], {x, -0.4, 1.5}, AspectRatio -> Automatic, AxesOrigin -> {0, 0}]


In Mathematica 8.0 this has been fixed (i.e. it will report inability to solve the equations.

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Wolfram Mathematica routinely confuses sums with integrals. This is just one example:

http://www.wolframalpha.com/input/?i=derivative+sum+2^(1/k)+from+0+to+x-1

$$\frac{\partial}{\partial x}\sum_0^{x-1} 2^{1/k} = 2^{1/k}$$