This calculator will try to find the infinite sum of arithmetic, geometric, power, and binomial series, as well as the partial sum, with steps shown (if possible). It will also check whether the series converges.

Our Series and Sum Calculator serves as an ideal tool for calculating the sum of different categories of sum and series. Whether you work with arithmetic or geometric sequences, our calculator will help you determine the sum quickly and efficiently.

## How to Use the Series and Sum Calculator?

### Input

Provide the general term of a series you need to find. Don't forget to provide the lower index (start value) and upper index (end value).

### Calculation

After inputting the required data, click the "Calculate" button.

### Result

The calculator will instantly provide the result.

## What Is a Mathematical Series?

A mathematical series is the summation of a sequence of terms given by a formula, that is, a certain rule. Each term of a sequence is typically generated based on the index (position) of that term in the sequence. For example, an infinite geometric series has the following form:

$$\sum_{i=1}^{\infty}\left(\frac{3}{5}\right)^i$$

Mathematical series have applications in various mathematical disciplines such as calculus, algebra, and number theory. Beyond theory, their usefulness extends to real-world areas of physics, engineering, and finance. Series are essential tools for analyzing and understanding mathematical principles, solving equations, and formulating predictive models.

**Arithmetic Series**An arithmetic series is the summation of a sequence of numbers in which each term is obtained by adding a constant value (common difference, denoted as $$$d$$$) to the previous term. The formula for the nth term of an arithmetic series is:

$$a_n=a_1+(n-1)d,$$where:

- $$$a_n$$$ is the nth term.
- $$$a_1$$$ is the first term.
- $$$d$$$ is the common difference.

The sum $$$S_n$$$ of the first $$$n$$$ terms of an arithmetic series can be calculated using the following formula:

$$S_n=\frac{n}{2}\left(2a_1+(n-1)d\right)$$For example, find the sum of the first $$$5$$$ terms of the arithmetic series with the first term $$$a_1$$$ equal to $$$3$$$ and a common difference $$$d$$$ equal to $$$2$$$.

Using the formula, we have:

$$S_5=\frac{5}{2}\left(2\cdot3+(5-1)\cdot2\right)=35$$The sum of the first $$$5$$$ terms is $$$35$$$.

**Geometric Series**A geometric series is the summation of a sequence in which each term is obtained by multiplying the previous term by a constant value. The formula for the nth term of a geometric series is

$$a_n=a_1r^{n-1},$$where:

- $$$a_n$$$ is the nth term.
- $$$a_1$$$ is the first term.
- $$$r$$$ is the common ratio.

The sum $$$S_n$$$ of the first $$$n$$$ terms of a geometric series can be calculated using the following formula:

$$S_n=\frac{a_1\left(1-r^n\right)}{1-r}$$For example, find the sum of the first $$$4$$$ terms of the geometric series with the first term $$$a_1$$$ equal to $$$2$$$ and a common ratio $$$r$$$ equal to $$$3$$$.

Using the formula, we have:

$$S_4=\frac{2\cdot\left(1-3^4\right)}{1-3}=80$$The sum of the first $$$4$$$ terms is $$$80$$$.

**Infinite Series**An infinite series is a series with an infinite number of terms. A common example is the geometric series. An infinite geometric series converges to a finite sum if the absolute value of the common ratio $$$r$$$ is less than $$$1$$$. In such cases, the sum of the infinite series can be calculated using the following formula:

$$S_{\infty}=\frac{a_1}{1-r}$$For example, find the sum of the infinite geometric series with $$$a_1=3$$$ and $$$r=\frac{1}{2}$$$.

Using the formula, we have:

$$S_{\infty}=\frac{3}{1-\frac{1}{2}}=6$$The sum of the infinite series is $$$6$$$.

## What Is the Difference Between a Sum and a Series?

The terms "sum" and "series" are related mathematical concepts, but they have distinct meanings. At its core, a sum is the result of adding a finite or infinite number of some terms. At the same time, a series is the summation of a finite or infinite sequence specified by some rule. For example, $$$\sum_{i=1}^5i=1+2+3+4+5$$$ is an arithmetic series, and its sum is $$$15$$$.

## Why Choose Our Series and Sum Calculator?

### Efficiency

Our calculator simplifies complex series calculations, saving you time and effort.

### Versatility

No matter if you're working with arithmetic, geometric, or other series, our calculator can handle many types of series easily.

### Accuracy

The calculator is created with accuracy in mind. Use it to exclude potential human errors.

### User-Friendly Interface

The user-friendly interface guarantees that individuals of any proficiency level can easily perform series calculations.

### FAQ

#### What is a series in math?

In mathematics, a series is the sum of a sequence of numbers or terms. Series can be finite or infinite. Series are used to analyze and model various mathematical phenomena, making them fundamental concepts.

#### What is the most famous mathematical series?

One of the most famous mathematical series is the geometric series. It is widely known for its simplicity and applications in various fields, including calculus and number theory. The geometric series consists of terms obtained by multiplying the previous term by a constant ratio.

#### How many types of series are there?

There are several types of mathematical series, but the two primary categories are:

- Arithmetic Series: Each term is obtained by adding a constant difference to the previous term.
- Geometric Series: Each term is generated by multiplying the previous term by a constant ratio.

Beyond these are power series, infinite series, and more, each with unique characteristics and applications in mathematics and science.

#### What is the Series and Sum Calculator with Steps?

The Series and Sum Calculator with Steps is an online mathematical tool designed to help you compute and understand various types of series. It provides solutions and answers for arithmetic, geometric, and other series, making it a valuable resource for both learning and practical applications.