Series Calculator
Analyze finite and infinite series with partial sums, convergence tests, graphs, tables, and step-by-step explanations.
Background
A series adds the terms of a sequence. This calculator helps students understand whether a series has a finite sum, how fast partial sums approach that value, and which convergence test explains the result.
How to use this calculator
- Choose general series, finite sum, geometric series, or convergence-test helper.
- Enter a term formula using n, or enter the geometric/test parameters.
- Click Analyze Series to see the sum, convergence status, graph, table, and explanation.
- Use quick examples to compare convergent, divergent, finite, geometric, and alternating series.
- Read the interpretation card to understand what the result means, not just the numeric value.
How this calculator works
- For finite sums, it evaluates every term and adds the running total.
- For general infinite series, it estimates partial sums and checks common recognizable patterns.
- For geometric series, it applies the exact convergence rule |r| < 1.
- For p-series and alternating series, it applies standard convergence-test logic.
- Graphs show both term values and cumulative partial sums so students can see whether totals settle down or keep growing.
Formula & Equations Used
Finite series: S_N = a_m + a_{m+1} + ... + a_N
Infinite series: Σ aₙ = lim S_N
Geometric series: a + ar + ar² + ...
Infinite geometric sum: S = a / (1 - r), |r| < 1
Finite geometric sum: S_N = a(1 - rᴺ) / (1 - r)
p-series rule: Σ 1/nᵖ converges when p > 1
Ratio test: L = lim |aₙ₊₁/aₙ|; converges if L < 1, diverges if L > 1
Example Problems & Step-by-Step Solutions
Example 1: p-series convergence
Determine whether Σ 1/n² converges.
This is a p-series with p = 2.
Because p > 1, the series converges.
Example 2: harmonic series divergence
Determine whether Σ 1/n converges.
This is a p-series with p = 1.
Because p ≤ 1, the harmonic series diverges.
Example 3: infinite geometric sum
Find the sum of 5 + 2.5 + 1.25 + ....
Here a = 5 and r = 0.5.
Since |r| < 1, S = a/(1-r) = 5/(1-0.5) = 10.
Example 4: alternating harmonic series
Analyze Σ (-1)ⁿ⁺¹/n.
The terms decrease toward 0 and alternate signs.
By the alternating series test, the series converges, but it does not converge absolutely.
Series concepts students often mix up
- Sequence vs. series: a sequence lists terms; a series adds them.
- Terms approaching zero is necessary, not sufficient: 1/n → 0, but Σ 1/n still diverges.
- Partial sum vs. infinite sum: a partial sum stops at N; an infinite sum depends on the limit as N grows.
- Conditional vs. absolute convergence: alternating series can converge even when the positive-term version diverges.
- Geometric series: the ratio test is about the size of r, not just whether terms look smaller at first.
FAQs
What is a series?
A series is the sum of terms from a sequence, such as a₁ + a₂ + a₃ + ....
What does it mean for a series to converge?
A series converges if its partial sums approach a finite value as more and more terms are added.
What does it mean for a series to diverge?
A series diverges if its partial sums do not approach a finite value.
Is this the same as a Taylor Series Calculator?
No. A Taylor Series Calculator expands functions into power series. This Series Calculator focuses on finite sums, infinite sums, convergence behavior, and partial sums.
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