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:

    “It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
    —G.C. (Georg Christoph)

    “No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
    —André Breton (1896–1966)

    “Every day the fat woman dies a series of small deaths.”
    —Shelley Bovey, U.S. author. Being Fat Is Not a Sin, ch. 1 (1989)