indefinite sum definition

Questions
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1

Hello, I am new to the Mathematica, so please tolerate with my naive question. Here I have a question about the indefinite summation, e.g.

input: Sum[i^3, i]
Output: 1/4(-1+i)^2*i^2

If I let i ranges from 1 to 1, e.g. {i,1,1}, the summation is 1. But according to the output expression, the result is 0. Why is that?

Thank you.

2

Your input Sum[i^3, i] is similar to an indefinite integral, which is defined only up to an additive constant. The output is correct in the following sense:

In[538]:= a[i_] = Sum[i^3, i];
a[i + 1] - a[i] // Simplify

Out[539]= i^3