When does an infinite geometric series converge?

An infinite geometric series converges (has a finite sum) only when the absolute value of the common ratio is less than 1 (|r| < 1). The sum is S∞ = a/(1-r). If |r| ≥ 1, the series diverges.

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Q: What is an arithmetic series?

An arithmetic series is the sum of terms where each term increases by a constant value (common difference). Formula: Sₙ = n/2 × (2a + (n-1)d) or Sₙ = n/2 × (first + last), where a is the first term, d is the common difference, and n is the number of terms.

Q: What is a geometric series?

A geometric series is the sum of terms where each term is multiplied by a constant ratio. Finite sum: Sₙ = a(1-rⁿ)/(1-r). Infinite sum (|r| < 1): S∞ = a/(1-r), where a is the first term and r is the common ratio.

Q: What is the difference between a sequence and a series?

A sequence is an ordered list of numbers (e.g., 1, 2, 3, 4, 5), while a series is the SUM of those numbers (e.g., 1 + 2 + 3 + 4 + 5 = 15). Sequences list terms; series add them.

Q: What is sigma notation?

Sigma notation (Σ) is a compact way to write series. The expression Σᵢ₌₁ⁿ aᵢ means 'sum all terms aᵢ from i=1 to i=n'. For example, Σᵢ₌₁⁵ i = 1+2+3+4+5 = 15.

Q: What is the sum of first n natural numbers?

The sum of first n natural numbers is n(n+1)/2. For example: 1+2+3+...+100 = 100×101/2 = 5050. This is a special case of an arithmetic series.

Q: What is a harmonic series?

The harmonic series is 1 + 1/2 + 1/3 + 1/4 + ... = Σ(1/n). Despite terms approaching zero, this series diverges (grows infinitely). It grows very slowly, roughly as ln(n) + γ (Euler's constant).

Calculate the sum of arithmetic, geometric, and special series. Get step-by-step solutions with nth term, convergence analysis, and term expansion.

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➕

Arithmetic Series Sum

Formula

Sₙ = n/2 × (2a + (n-1)d)

📖 Definition

An arithmetic series is the sum of terms where each term increases by a constant difference. Examples: 2+5+8+11 (d=3) or 10+7+4+1 (d=-3).

First Term (a)

Common Difference (d)

Number of Terms (n)

Sum of Series (Sₙ)

Sₙ = 5/2 × (2×1 + (5-1)×2) = 25

First Term

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Last Term

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nth Term Formula

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📚 Series Formulas

Complete reference for arithmetic, geometric, and special series formulas.

➕ Arithmetic Series

Sₙ = n/2 × (2a + (n-1)d)

Also: Sₙ = n/2 × (first + last). a = first term, d = common difference, n = terms.

✖️ Geometric Series (Finite)

Sₙ = a(1 - rⁿ)/(1 - r)

a = first term, r = common ratio (r ≠ 1), n = number of terms.

∞ Geometric Series (Infinite)

S∞ = a/(1 - r)

Only converges when |r| < 1. Otherwise, series diverges to infinity.

🔢 Sum of Natural Numbers

Σn = n(n+1)/2

1 + 2 + 3 + ... + n. Also called triangular numbers.

² Sum of Squares

Σn² = n(n+1)(2n+1)/6

1² + 2² + 3² + ... + n². Used in statistics (variance).

³ Sum of Cubes

Σn³ = [n(n+1)/2]²

1³ + 2³ + 3³ + ... + n³ = (Σn)². Sum of cubes equals square of sum.

🔢 Special Series Reference

Common Series & Convergence

Series Name	Expression	Sum / Behavior	Converges?
Natural Numbers	1 + 2 + 3 + ... + n	n(n+1)/2	Finite
Sum of Squares	1² + 2² + ... + n²	n(n+1)(2n+1)/6	Finite
Sum of Cubes	1³ + 2³ + ... + n³	[n(n+1)/2]²	Finite
Geometric (\|r\|<1)< /td>	a + ar + ar² + ...	a/(1-r)	Yes
Geometric (\|r\|≥1)	a + ar + ar² + ...	Grows to ∞	No
Harmonic Series	1 + 1/2 + 1/3 + ...	≈ ln(n) + γ	Diverges
p-Series (p>1)	Σ 1/nᵖ	Finite value	Yes
p-Series (p≤1)	Σ 1/nᵖ	Grows to ∞	No

❓ Frequently Asked Questions

Common questions about series and summation answered.

What is a series in mathematics?+

A series is the sum of the terms of a sequence. For example, 1 + 2 + 3 + 4 + 5 = 15 is the series formed by summing the first five natural numbers. Series can be finite or infinite.

What is the difference between a sequence and a series?+

A sequence is an ordered list of numbers (1, 2, 3, 4, 5), while a series is the SUM of those numbers (1 + 2 + 3 + 4 + 5 = 15). Sequences list; series add.

What is an arithmetic series?+

An arithmetic series sums terms with a constant difference between consecutive terms. Formula: Sₙ = n/2 × (2a + (n-1)d), where a is first term, d is common difference, n is term count.

What is a geometric series?+

A geometric series sums terms where each term is multiplied by a constant ratio. Finite: Sₙ = a(1-rⁿ)/(1-r). Infinite (|r|<1): S∞=a/(1-r).

What is sigma notation?+

Sigma notation (Σ) compactly writes series. Σᵢ₌₁ⁿ aᵢ means "sum aᵢ from i=1 to n". Example: Σᵢ₌₁⁵ i = 1+2+3+4+5 = 15.

When does an infinite series converge?+

An infinite series converges when its partial sums approach a finite limit. For geometric series, this requires |r| < 1. The harmonic series diverges despite terms approaching zero.

What is the sum of first n natural numbers?+

The sum is n(n+1)/2. Example: 1+2+...+100 = 100×101/2 = 5050. This formula was famously discovered by young Gauss.

What is a harmonic series?+

The harmonic series is 1 + 1/2 + 1/3 + 1/4 + ... = Σ(1/n). Despite terms approaching zero, it diverges (grows infinitely), approximately as ln(n) + 0.5772 (Euler's constant).