No. On the globe, the main change is tilt, which depends on the observer's position, and time, not the distance to the stars.You seem to have come up with a number for "very very far", as that's one of the inputs necessary for getting a number out.
In empirical practice, the width of Earth's orbit needed to be utilized to discover stellar parallax with astronomical instruments. That means, apart from tilt, the sky is pretty much the same (for household purposes) everywhere on Earth.As your parenthetical "infinitely" indicates, you're justified in just taking the limit as the number you chose tends to infinity, and that leads to a tilt of 0⁰ in the limit (which is well behaved). You were right at the outset, the adding of numbers has fuzzied things.
"The finite is annihilated in the presence of the infinite, and becomes a pure nothing. " — Blaise Pascal, On the Wager for God’s Existence"All finite things are but nothing to the infinite" - Epifatphilus, famous non-existent non-greek non-philosopher.