Sequences and series

• Is 1/3 Equal to 0.333...?
  Yes. The repeating decimal 0.333... means the limit of 0.3, 0.33, 0.333, and so on, and that limit is exactly 1/3.
• How to Solve Integrals Using Series
  Use a power series to replace a difficult integrand with simpler terms, then integrate term by term after checking convergence on the interval.
• How to Find Closed Forms for Summations
  Learn practical ways to find finite-sum formulas, including pattern recognition, telescoping, index shifts, and a worked example.
• What Is an Example of an Unbounded Set?
  The natural numbers are an example of an unbounded set: no finite real number is an upper bound for all of their elements.
• Can a Double Sequence Have Different Iterated Limits?
  A double sequence can have different iterated limits: for aₘₙ = m/(m+n), the two orders give 0 and 1, so no joint limit exists.
• 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.
• Can a Convergent Sequence Be Unbounded?
  No: every convergent real sequence is bounded. Bound its tail using the limit, then take a maximum over the finitely many earlier terms.
• Find the Supremum, Infimum, lim sup, and lim inf of 2 − 1/n
  For n ≥ 1, the sequence 2 − 1/n has infimum and minimum 1, supremum 2, and both lim sup and lim inf equal to 2.