Set gives an unexpected error message (Set::partw)

Question

Here is a minimal example:

s = 1 /(a + b[3]) + 1 /(2 a + b[3]);
s[[1]] = Sum[s[[1]] /. b[3] -> k, {k, 3}]
s[[4]]
s[[4]] = Sum[s[[4]] /. b[3] -> k, {k, 3}]

Output:

Out[143]= 1/(1 + a) + 1/(2 + a) + 1/(3 + a)
Out[144]= 1/(2 a + b[3])

Set::partw: Part 4 of (1/(1+a)+1/(2+a)+1/(3+a))+1/(2 a+b[3]) does not exist. >>

Out[145]= 1/(1 + 2 a) + 1/(2 + 2 a) + 1/(3 + 2 a)

I could only make the error occur when using Set. As the above code shows, both s[[4]] and Sum[s[[4]] /. b[3] -> k, {k, 3}] are well defined.

I have been struggling with similar errors for days, can someone please explain exactly what is happening behind the scenes?

Answer 1

The reason this error occurs is because Set and Part have the Attribute HoldFirst, while functions like Position, which was used to get the position 4, does not have HoldFirst.

To verify, the line

s[[1]] = Sum[s[[1]] /. b[3] -> k, {k, 3}]

adds a parenthesis which does not get evaluated, so

(s//Hold)[[1,4]]
Out[1] = Part::partw: "Part 4 of (1/(1+a)+1/(2+a)+1/(3+a))+1/(2\a+b[3]) does not exist."

where as using Evaluate will distribute the parenthesis (effectively distributing the function Plus)

(s//Evaluate //Hold)[[1,4]]
1/(2 a + b[3])

So as pointed out by @andre executing s=s before s[[4]]= ... would solve the problem.

I thank @andre and @rasher for their contributions.