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Series & Sequences

Convergence testing, closed-form sums, Fourier series, and power series — exact symbolic results with every step explained.

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What Zeretis can solve

Zeretis handles both finite and infinite series — determining convergence, computing exact sums where they exist, and generating Fourier and power series expansions.

Convergence tests

Ratio test, root test, comparison test, integral test, alternating series test — applied and explained.

Arithmetic sequences

General term, partial sums, common difference identification.

Geometric series

Finite and infinite sums, convergence condition |r| < 1, exact closed forms.

Power series

Radius of convergence, interval of convergence, term-by-term differentiation and integration.

Fourier series

Fourier coefficients for periodic functions, sine and cosine series.

Telescoping series

Partial fraction decomposition to find exact sums of telescoping series.

How to type series problems

Convergence test
does the series sum of 1/n^2 converge
→ Converges (p-series with p=2 > 1, sum = π²/6)
Geometric series
sum of (1/2)^n from n=0 to infinity
→ 2 (geometric series, r = 1/2)
Power series radius
radius of convergence of sum of n! * x^n
→ R = 0 (diverges for all x ≠ 0)
Arithmetic sequence
sum of first 100 natural numbers
→ 5050

Best practices

  • For convergence, phrase it as "does the series ... converge" — the solver picks the best test
  • Specify the index and starting value: "sum from n=1 to infinity of..."
  • For Fourier series: "Fourier series of f(x) = x on [-π, π]"
  • Power series: "find the power series for sin(x) around 0" gives the Maclaurin series
  • The solver shows which convergence test was applied and why
📐Algebra ∫Calculus 🔀Differential Equations 🔢Linear Algebra 📏Trigonometry

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