Convergent Series - Examples of Convergent and Divergent Series

Examples of Convergent and Divergent Series

  • The reciprocals of the positive integers produce a divergent series (harmonic series):
  • Alternating the signs of the reciprocals of positive integers produces a convergent series:
  • Alternating the signs of the reciprocals of positive odd integers produces a convergent series (the Leibniz formula for pi):
  • The reciprocals of prime numbers produce a divergent series (so the set of primes is "large"):
  • The reciprocals of triangular numbers produce a convergent series:
  • The reciprocals of factorials produce a convergent series (see e):
  • The reciprocals of square numbers produce a convergent series (the Basel problem):
  • The reciprocals of powers of 2 produce a convergent series (so the set of powers of 2 is "small"):
  • Alternating the signs of reciprocals of powers of 2 also produce a convergent series:
  • The reciprocals of Fibonacci numbers produce a convergent series (see ψ):

Read more about this topic:  Convergent Series

Famous quotes containing the words examples of, examples and/or series:

    “Histories are more full of examples of the fidelity of dogs than of friends.”
    —Alexander Pope (1688–1744)

    “In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
    —Michel de Montaigne (1533–1592)

    “History is nothing but a procession of false Absolutes, a series of temples raised to pretexts, a degradation of the mind before the Improbable.”
    —E.M. Cioran (b. 1911)