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When do I use GenerateConditions and Assumptions with Integrate or Sum?

Integrate and Sum evaluations return generic solutions. These are usually correct for general cases, but may not apply for specific parameter values (see Generic and Non-Generic Cases).

For example, this summation is unbounded if x is greater than or equal to 1:

In[1]:= Sum[x^n, {n, 0, Infinity}]

Out[1]=  1 / (1 - x)  

The GenerateConditions->True option tells the function to state when the solution is valid.

Now we confirm that the result applies only for Abs[x]<1:

In[2]:= Sum[x^n, {n, 0, Infinity}, GenerateConditions -> True]

Out[2]=  ConditionalExpression[1/(1 - x), Abs[x] < 1]  

If any condition is known already, the Assumptions option can be used to tell Sum about it. This gives a simple output suitable for later use in the code. The explicit condition used here will need to be remembered when the result is used:

In[3]:= Sum[x^n, {n, 0, Infinity}, Assumptions -> {-1 < x && x < 1}]

Out[3]= 1/(1 - x)    

Assumptions can also be passed using the Assuming function or $Assumptions:

In[4]:= Integrate[1/(x + a), {x, 0, 1}]

Out[4]= ConditionalExpression[-Log[a] + Log[1 + a],
   Re[a] > 0 || Re[a] < -1 || NotElement[a, Reals]

In[5]:= Assuming[a > 0, Integrate[1/(x + a), {x, 0, 1}]]

Out[5]= Log[1 + 1/a]

In[6]:= $Assumptions = a > 0;
        Integrate[1/(x + a), {x, 0, 1}]

Out[7]= Log[1 + 1/a]

This resets $Assumptions to its default:

In[8]:= $Assumptions =. ;  

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