sum of series calculator

Sum of Series Calculator

Select Series Type: First Term (a): Common Difference / Ratio (d or r): Number of Terms (n):

Sum: 0

🔢 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

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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.