Series Sum Calculator

Calculate the sum of arithmetic, geometric, power series, and more. Visualize partial sums and convergence.

Summation notation: ∑ aₙ from n = start to end.
Enter the term as an expression in n (or your chosen variable).

Expression in n (or custom variable). Use ^ for exponent, * for multiplication, functions: sin, cos, exp, log, sqrt, etc.
1/n (Harmonic)
1/n² (p-series)
(-1)ⁿ⁺¹/n
n²
2ⁿ
(½)ⁿ
sin(1/n)
1/(n(n+1))
Single letter, avoid e, pi, i (math constants).
Step > 1 skips terms, computing a subsequence sum.

Understanding Series Summation

A series is the sum of the terms of a sequence. We denote the partial sum SN = ∑_{n=n₀}^{N} aₙ. If the limit as N→∞ exists (finite), the series converges to that limit; otherwise it diverges.

Fundamental Definition:

S = lim_{N→∞} S_N = lim_{N→∞} ∑_{n=n₀}^{N} a_n

If the limit does not exist or is infinite, the series diverges.

Common Series & Closed-Form Formulas

Series Partial Sum SN Infinite Sum (if convergent)
Arithmetic: ∑_{k=1}^{N} k N(N+1)/2 diverges
Geometric: ∑_{k=0}^{N} rk (r≠1) (1 - r^{N+1})/(1 - r) 1/(1 - r) for |r| < 1
p-series: ∑_{k=1}^{∞} 1/kp — converges if p > 1
Alternating harmonic: ∑ (-1)^{k+1}/k — ln 2
Basel: ∑ 1/k² — π²/6
Exponential: ∑ xk/k! — ex

Convergence Tests (for positive-term series)

Test Condition for Convergence Example
n-th term test If lim aₙ ≠ 0, series diverges (necessary condition) ∑ n/(n+1) diverges
Geometric series Converges iff |r| < 1 ∑ (1/2)ⁿ converges to 2
p-series Converges iff p > 1 ∑ 1/n¹·⁵ converges
Ratio test lim |aₙ₊₁/aₙ| = L < 1 → converges; L > 1 → diverges ∑ n/2ⁿ converges (L=1/2)
Root test lim sup |aₙ|^{1/n} = L < 1 → converges ∑ (n/3ⁿ) converges (L=1/3)
Integral test ∫ f(x) dx converges ↔ ∑ f(n) converges (f positive decreasing) ∑ 1/(n ln n) diverges (integral diverges)
Alternating series test bₙ decreasing → 0 ⇒ ∑ (-1)ⁿ bₙ converges ∑ (-1)ⁿ/n converges (conditional)

Absolute vs Conditional Convergence

  • Absolute convergence: ∑ |aₙ| converges ⇒ ∑ aₙ converges.
  • Conditional convergence: ∑ aₙ converges but ∑ |aₙ| diverges (e.g., alternating harmonic series).

Visualizing Convergence

The graph of partial sums SN versus N helps identify convergence: if the curve approaches a horizontal asymptote, the series converges to that value. If it grows without bound or oscillates without settling, it diverges.

Calculator Features & Limitations

  • Uses math.js for safe evaluation of term expressions.
  • Computes finite sums up to N (you can set N large to approximate infinite series).
  • Displays partial sums table and convergence graph.
  • Note: Very large N (>10⁵) may be slow; for alternating series, ensure step size captures all terms.

Frequently Asked Questions

A sequence is an ordered list of numbers (a₁, a₂, a₃,…). A series is the sum of the terms of a sequence (a₁ + a₂ + a₃ + …).

This calculator computes finite sums. For infinite series, you can approximate by taking a large N and observing if partial sums stabilize. Use convergence tests for rigorous analysis.

Yes, you can specify any single-letter index variable (e.g., k, i, m). The term expression should use that variable.

Plotting partial sums helps visualize convergence: if the curve flattens to a horizontal line, the series converges.

The calculator will still compute the sum up to the given N, but the graph will show unbounded growth or oscillation. Use the tests above to confirm divergence.