Sum of Series Calculator
🔢 Sum of Series Calculator – Quickly Find the Sum of a Sequence or Series
A Sum of Series Calculator is a tool that calculates the total sum of a sequence or mathematical series, whether finite or infinite. It supports arithmetic, geometric, and sometimes even custom formula-based series.
📘 What Is a Series?
A series is the sum of the terms of a sequence. Sn=a1+a2+a3+⋯+anS_n = a_1 + a_2 + a_3 + \cdots + a_nSn=a1+a2+a3+⋯+an
Series are used in algebra, calculus, finance, and physics to compute running totals, estimate areas, or solve real-world problems.
🔍 Types of Series You Can Calculate
| Series Type | Formula | Conditions |
|---|---|---|
| Arithmetic Series | Sn=n2(2a+(n−1)d)S_n = \frac{n}{2}(2a + (n – 1)d)Sn=2n(2a+(n−1)d) | Constant difference ddd |
| Geometric Series | Sn=a⋅1−rn1−rS_n = a \cdot \frac{1 – r^n}{1 – r}Sn=a⋅1−r1−rn (finite) | Common ratio r≠1r \ne 1r=1 |
| S=a1−rS = \frac{a}{1 – r}S=1−ra (infinite) if ( | r | |
| Harmonic Series | ∑n=1N1n\sum_{n=1}^{N} \frac{1}{n}∑n=1Nn1 | No closed formula, grows slowly |
| Custom Series | Based on general term like nn+1\frac{n}{n+1}n+1n, n2n^2n2, etc | Handled by symbolic calculators |
🧠 Example Calculations
🔹 Finite Arithmetic Series
∑n=110(3n+1)⇒Use arithmetic rules\sum_{n=1}^{10} (3n + 1) \Rightarrow \text{Use arithmetic rules}n=1∑10(3n+1)⇒Use arithmetic rules
🔹 Finite Geometric Series
∑n=042n=1+2+4+8+16=31\sum_{n=0}^{4} 2^n = 1 + 2 + 4 + 8 + 16 = 31n=0∑42n=1+2+4+8+16=31
🔹 Infinite Geometric Series
∑n=0∞(13)n=11−13=32\sum_{n=0}^{\infty} \left(\frac{1}{3}\right)^n = \frac{1}{1 – \frac{1}{3}} = \frac{3}{2}n=0∑∞(31)n=1−311=23
💡 Features of a Good Sum of Series Calculator
- Accepts closed-form or general term input
- Supports finite & infinite series
- Graphs series or terms (optional)
- Gives step-by-step simplification (in tools like Symbolab or WolframAlpha)
- Tells whether a series converges or diverges
❓ FAQs – Sum of Series Calculator
🔹 What is the difference between a sequence and a series?
A sequence is a list of numbers. A series is the sum of that list.
🔹 Can this calculator solve both finite and infinite series?
Yes — depending on the tool. Some tools give exact answers for both.
🔹 What if the series doesn’t converge?
For divergent infinite series, the calculator typically reports “no sum” or “diverges.”
🔹 How can I enter a custom series formula?
Most calculators accept n as the index. Example:
- an=nn+1a_n = \frac{n}{n+1}an=n+1n
- Series: ∑n=110nn+1\sum_{n=1}^{10} \frac{n}{n+1}∑n=110n+1n
🔹 Is it the same as “Infinite Sum Calculator”?
The infinite sum calculator is a subset of the sum of series calculators — specific to converging infinite series.