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
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