Calculators
Harmonic Sequence Calculator
Calculate the nth term and finite sum of a harmonic sequence from an arithmetic progression of denominators, privately in your browser.
Calculate reciprocals of an arithmetic progression.
A harmonic sequence consists of reciprocals of an arithmetic sequence: aₙ = 1 / [b₁ + (n − 1)d], where b₁ is the first denominator and d is the difference between denominators. For b₁ = 1 and d = 1, the first four terms are 1, 1/2, 1/3, and 1/4; their sum is approximately 2.08333333333. The calculator adds the requested terms numerically and previews the first ten. Enter denominators rather than the first reciprocal term: a first term of 1/2 means b₁ = 2. Negative denominators and differences are allowed, but any zero denominator in the requested range makes the sequence undefined. A zero difference gives a constant sequence when b₁ is nonzero. The term count must be a whole number from 1 to 10,000. Signed decimals and scientific notation are accepted. Browser floating-point arithmetic can lose precision near zero or with large magnitudes; displayed values use up to 12 significant digits. This finite sum is not a convergence test for an infinite series. All calculations stay in your browser.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What is a harmonic sequence?
Its terms are reciprocals of an arithmetic progression. With denominators 1, 2, 3, and 4, the terms are 1, 1/2, 1/3, and 1/4.
Which first value should I enter?
Enter the first denominator b₁, not its reciprocal. For a sequence beginning 1/2, 1/5, 1/8, enter 2 as the first denominator and 3 as the difference.
Can denominators be negative or zero?
Negative denominators and differences are allowed. If any requested denominator is zero, the calculator reports the affected term instead of returning an undefined result.
Does this calculate an infinite sum?
No. It calculates a finite sum for 1 to 10,000 terms using browser floating-point arithmetic, displaying up to 12 significant digits. The standard infinite harmonic series diverges. Your inputs remain in your browser.