Series vs. sequence (and why Σ matters)

A sequence is a list of numbers in order: a₁, a₂, a₃, … A series is what you get when you add the terms of a sequence: Sₙ = a₁ + a₂ + … + aₙ. When you see the sigma symbol (Σ), it’s simply shorthand for “sum these terms.”

Example: the sequence 3, 5, 7, 9, … is arithmetic (each term increases by 2). The corresponding series for the first n terms is: Sₙ = 3 + 5 + 7 + … (n terms). In many real problems, you don’t just want the nth term — you want the total up to n: total distance, total cost, total points, total savings, total pixels, or total error. That’s why series show up everywhere: finance, physics, computer science, and even “how many minutes have I spent procrastinating this week?” (yes, a series can model that too).

• Arithmetic series: terms change by adding a constant difference d.
• Geometric series: terms change by multiplying by a constant ratio r.
• Harmonic series: terms are reciprocals: 1/k. It grows without bound, but slowly.

The calculator focuses on these because they’re the most common “pattern series” taught in algebra, pre‑calculus, and calculus — and they’re the most shareable for quick checks in study groups. Next, let’s break down each one with formulas, intuition, and examples you can reuse.

Arithmetic series (constant difference)

An arithmetic sequence looks like: a₁, a₁ + d, a₁ + 2d, a₁ + 3d, … The nth term is: aₙ = a₁ + (n − 1)d. The arithmetic series is the sum of the first n terms: Sₙ = a₁ + (a₁ + d) + … + (a₁ + (n − 1)d).

There are two classic ways to compute the arithmetic sum. The fastest (and most famous) is the “pairing trick” credited to young Gauss. Write the series forwards and backwards:

Sₙ = a₁ + (a₁ + d) + … + aₙ Sₙ = aₙ + (aₙ − d) + … + a₁

Add them term-by-term. Each pair becomes (a₁ + aₙ), and there are n pairs: 2Sₙ = n(a₁ + aₙ) → so: Sₙ = n(a₁ + aₙ)/2. Substitute aₙ = a₁ + (n − 1)d to get the popular “all-in-one” form: Sₙ = n/2 · (2a₁ + (n − 1)d).

Suppose you start with 3 points on day 1 and add 2 points each day. After 10 days: a₁ = 3, d = 2, n = 10. The 10th term: a₁₀ = 3 + (10 − 1)·2 = 21. The total points in 10 days: S₁₀ = 10/2 · (2·3 + 9·2) = 5 · (6 + 18) = 120. This is exactly the kind of “total accumulation” question where arithmetic series are perfect.

Geometric series (constant ratio)

A geometric sequence looks like: a₁, a₁r, a₁r², a₁r³, … The nth term is: aₙ = a₁ · r^(n − 1). The sum of the first n terms is the finite geometric series: Sₙ = a₁ + a₁r + a₁r² + … + a₁r^(n − 1).

The finite sum has a beautiful closed form. Multiply the sum by r and subtract:

Sₙ = a₁ + a₁r + a₁r² + … + a₁r^(n − 1) rSₙ = a₁r + a₁r² + … + a₁r^n Subtract: Sₙ − rSₙ = a₁ − a₁r^n

Factor: Sₙ(1 − r) = a₁(1 − r^n), so when r ≠ 1: Sₙ = a₁(1 − r^n)/(1 − r). If r = 1, every term equals a₁ and the sum is just Sₙ = n·a₁.

If a video goes viral and the views multiply by 1.5 each day starting at 200 views: a₁ = 200, r = 1.5, n = 7. Then: S₇ = 200(1 − 1.5^7)/(1 − 1.5). Because the denominator is negative, the value is still positive overall. The calculator prints the numeric result and the substitution steps so you can see where the signs go.

Infinite geometric series (convergence)

Some geometric series keep going forever: S = a₁ + a₁r + a₁r² + …. The key question is: does the sum settle to a finite number, or does it blow up? The answer depends on the size of r. If |r| < 1, the powers of r shrink toward zero, and the series converges. In that case: S = a₁/(1 − r). If |r| ≥ 1, the terms don’t shrink fast enough, and the series diverges.

This is more than a textbook trick — it models repeating decimals, discounted cash flows, and “infinite process” problems. A classic example is: 1/2 + 1/4 + 1/8 + … which sums to 1. Here a₁ = 1/2, r = 1/2, so: S = (1/2)/(1 − 1/2) = 1.

Harmonic series (diverges slowly)

The harmonic series is: Hₙ = 1 + 1/2 + 1/3 + … + 1/n. Unlike geometric series with |r| < 1, the harmonic series does not converge as n → ∞ — it grows without bound. But it grows very slowly, which makes it feel “almost convergent” for small n.

For intuition, compare it to the natural logarithm. A famous approximation is: Hₙ ≈ ln(n) + γ where γ is the Euler–Mascheroni constant (about 0.57721…). Our calculator computes the exact partial sum by direct addition, and also shows the ln(n) + γ approximation so you can see why the growth is slow.

How the calculator decides what to show

Each series type produces slightly different outputs:

• Arithmetic: nth term aₙ, partial sum Sₙ, and the first few terms preview.
• Geometric: finite sum Sₙ; optionally infinite sum S if |r| < 1; plus nth term.
• Harmonic: partial sum Hₙ and approximation ln(n)+γ.

We also include a small “convergence meter.” It’s not a rigorous proof tool — it’s a visual hint that’s good for teaching. For example, an infinite geometric series with |r| = 0.1 is strongly convergent (meter near 100). With |r| = 0.95 it’s convergent but slow (meter closer to the middle). The harmonic series is always set near 0 on this meter because it diverges as n increases.

Frequently Asked Questions

• What’s the difference between a series and a sequence?

  A sequence is a list of terms (a₁, a₂, …). A series is the sum of those terms. Most homework uses Σ notation to represent a series (a sum).

• How do I know if a geometric series converges?

  For an infinite geometric series, it converges only if |r| < 1. If that condition holds, the sum is a₁/(1 − r).

• What if r = 1 in a geometric series?

  Then every term is equal to a₁ and the finite sum is Sₙ = n·a₁. Infinite with r = 1 diverges (it grows without bound unless a₁ = 0).

• Does the harmonic series ever converge?

  Not as n → ∞. The partial sums grow roughly like ln(n) + γ, meaning it diverges very slowly. For small n it might look “stable,” but mathematically it keeps increasing.

• Can I use this for Taylor or Fourier series?

  This page focuses on the most common “parameterized” series types. For specialized expansions, use the dedicated tools (Taylor Series Calculator / Fourier Series Calculator) linked below.

• Are results stored or sent to a server?

  No. Everything runs in your browser. If you click “Save Result,” we store only the summary text in your localStorage so you can compare later on the same device.

Fast workflow (best for virality)

Want the fastest, most shareable result? Use this 3‑step flow:

• Pick type: Arithmetic / Geometric / Harmonic
• Enter numbers: a₁ and d (or r), plus n
• Tap Calculate: then press “Copy” and paste the Σ steps anywhere

The saved history list is also great for content: you can show how Sₙ changes as n increases (a popular “growth curve” TikTok/shorts idea).

Common mistakes (avoid these)

• Arithmetic: don’t forget (n − 1) when computing aₙ.
• Geometric: if r is close to 1, sums can get big fast — use enough precision.
• Infinite geometric: only valid when |r| < 1.
• Harmonic: diverges — don’t try to use an “infinite sum” formula.

If you’re stuck, compute the first 4–6 terms by hand. If the pattern matches what you picked, the formula will match too.