Two bugs in Rev 6975

[Snorre]

So your first problem remains. (How did the iteration variable pop up to the result?)
I'm not sure if it is related to: I always try to avoid to calculate differentials (as in taylor series expansion) of sums -- e.g. diff(sum(x/k²+1,k=1..∞),x) vs. sum(diff(x/k²+1,x),k=1..∞), where both should be equal, but due to evaluation order from inner to outer the first expression results in eyebrow-lifting ∂∞/∂x=0.
Maybe you should better do a sum(taylor(x^(c+1),c=5),x=1..b) which is also a taylor polynom, but won't help much as without a concrete b the sum cannot be further simplified.