Infinite series

• Does a Countably Infinite Sum of Zeros Equal Zero?
  A countable series whose every term is zero has sum zero: every partial sum is zero, so its limit is zero, regardless of enumeration.
• What Is the Sum of the Alternating Reciprocal-Factorial Series?
  The alternating reciprocal-factorial series sums to 1 − 1/e when it starts with 1/1!. See the indexing variants and how to bound errors.
• How Do You Differentiate a Double Summation?
  Differentiate finite double sums term by term; for infinite sums, check uniform convergence of the derivative series before interchanging differentiation and summation.
• How Do Limits Resolve Zeno’s Achilles Paradox?
  Zeno’s Achilles paradox is resolved mathematically by showing that infinitely many shrinking time intervals sum to a finite limit, when motion is modeled continuously.