Sum of Series Calculator

Understanding Sum of Series

What is a Sum of Series?

A sum of series is the result of adding up all the terms in a sequence of numbers. In mathematics, we often deal with two main types of series: arithmetic series and geometric series.

Arithmetic Series

An arithmetic series is a sequence of numbers where the difference between each consecutive term is constant. This constant difference is called the common difference.

Formula for the Sum of an Arithmetic Series

\[S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}(2a + (n-1)d)\]

Where:

• \(S_n\) is the sum of the series
• \(n\) is the number of terms
• \(a_1\) is the first term
• \(a_n\) is the last term
• \(a\) is the first term
• \(d\) is the common difference

Example of an Arithmetic Series

Let's consider the arithmetic series: 2, 5, 8, 11, 14

Here, a = 2, d = 3, and n = 5

Using the formula:

\[S_5 = \frac{5}{2}(2(2) + (5-1)(3)) = \frac{5}{2}(4 + 12) = \frac{5}{2}(16) = 40\]

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Formula for the Sum of a Geometric Series

For r ≠ 1: \[S_n = \frac{a(1-r^n)}{1-r}\]

For r = 1: \[S_n = an\]

Where:

• \(S_n\) is the sum of the series
• \(a\) is the first term
• \(r\) is the common ratio
• \(n\) is the number of terms

Example of a Geometric Series

Let's consider the geometric series: 3, 6, 12, 24

Here, a = 3, r = 2, and n = 4

Using the formula:

\[S_4 = \frac{3(1-2^4)}{1-2} = \frac{3(1-16)}{-1} = \frac{3(-15)}{-1} = 45\]

Understanding the sum of series is crucial in many areas of mathematics and its applications, including finance (compound interest), physics (motion calculations), and computer science (algorithm analysis).