Σ Sum of Series Calculator

Arithmetic Series: a + (a+d) + (a+2d) + ... + (a+(n-1)d)

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Geometric Series: a + ar + ar² + ar³ + ... + arⁿ⁻¹

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Infinite Series: ∑(n=1 to ∞) aₙ

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Custom Series: Define Your Own Terms

Supports: +, -, *, /, ^, sin(), cos(), tan(), ln(), sqrt(), abs(), factorial notation (!)
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Understanding Series and Sequences

A series is the sum of terms in a sequence. Understanding different types of series and their convergence properties is fundamental in calculus, analysis, and many applications in science and engineering. Series such as arithmetic, geometric, and power series each have unique characteristics and formulas that make them useful for modeling and solving real-world problems.

Working with series often involves calculating partial sums or sums over combinational selections of terms, which can become complex for large or variable data sets. To assist in these computations, the Combination Sum Calculator offers a versatile tool for efficiently evaluating sums of various combinations and sequences.

Whether you are studying convergence to a limit, investigating series behavior, or solving engineering problems, using an interactive calculator designed for combination sums can save time and enhance accuracy. This tool empowers users to explore series with multiple constraints and summation patterns, making advanced mathematical concepts more approachable.

General Series: ∑n=1N aₙ = a₁ + a₂ + a₃ + ... + aₙ

Types of Series

📐 Arithmetic Series

Terms increase by constant difference

Sₙ = n/2 × (2a + (n-1)d)

Example: 2 + 5 + 8 + 11 + 14 (d=3)

📊 Geometric Series

Terms multiply by constant ratio

Sₙ = a(1-rⁿ)/(1-r) when r≠1

Example: 3 + 1.5 + 0.75 + 0.375 (r=0.5)

∞ Infinite Series

Sum of infinitely many terms

S∞ = a/(1-r) when |r|<1

Convergence: Critical for meaningful sums

🔧 Custom Series

User-defined term expressions

aₙ = f(n), any function of n

Examples: n², 1/n!, sin(n), 2ⁿ

Convergence and Divergence

🔎 Convergence Tests

Geometric Series Test:
∑arⁿ converges if |r| < 1, sum=a/(1-r)
P-Series Test:
∑1/nᵖ converges if p > 1, diverges if p ≤ 1
Ratio Test:
lim |aₙ₊₁/aₙ| < 1 → converges,> 1 → diverges
Alternating Series Test:
∑(-1)ⁿaₙ converges if aₙ↓0 and aₙ > 0

Famous Series Examples

Example 1: Geometric Series

Series: 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...

First term: a = 1
Common ratio: r = 1/2
Since |r| = 1/2 < 1, series converges
Sum = a/(1-r) = 1/(1-1/2) = 2

Example 2: Arithmetic Series

Series: 3 + 7 + 11 + 15 + 19 (first 5 terms)

First term: a = 3
Common difference: d = 4
Number of terms: n = 5
Sum = n/2 × (2a + (n-1)d) = 5/2 × (6 + 16) = 55

Example 3: P-Series

Series: ∑(1/n²) = 1 + 1/4 + 1/9 + 1/16 + ...

P-value: p = 2 > 1
Series converges
Exact sum = π²/6 ≈ 1.6449
Basel problem solution by Euler

Frequently Asked Questions

A sequence is an ordered list of numbers (like 1, 2, 3, 4...), while a series is the sum of the terms in a sequence (like 1+2+3+4+... = 10 for the first 4 terms). Think of a sequence as the individual terms and a series as their total.
Use convergence tests: For geometric series, check if |r| < 1. For p-series (∑1/nᵖ), it converges if p> 1. For other series, try the ratio test, root test, or comparison test. Our calculator automatically performs these tests for supported series types.
If |r| > 1, the series diverges (grows without bound). If r = 1, you get a + a + a + ..., which also diverges unless a = 0. If r = -1, you get a - a + a - a + ..., which oscillates and doesn't converge to a single value.
Yes! Use our Custom Series calculator to compute the first N terms of any series. This gives you partial sums that approximate the infinite sum. For convergent series, increasing N gives better approximations of the true infinite sum.
Series appear everywhere: compound interest (geometric series), signal processing (Fourier series), physics (Taylor series for approximations), economics (present value calculations), computer graphics (infinite series for curves), and probability (generating functions).

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