What is the sum?

Summation (Σ)

Summation, often denoted by the Greek capital letter sigma (Σ), is a concise way to express the addition of a sequence of terms. It's a fundamental operation in <a href="https://www.wikiwhat.page/kavramlar/mathematics">mathematics</a> and appears across diverse fields like <a href="https://www.wikiwhat.page/kavramlar/statistics">statistics</a>, <a href="https://www.wikiwhat.page/kavramlar/physics">physics</a>, and <a href="https://www.wikiwhat.page/kavramlar/computer%20science">computer science</a>.

Key Components:

• Index Variable: A variable (e.g., i, j, k) that represents the current term being added.
• Lower Limit: The starting value of the index variable.
• Upper Limit: The ending value of the index variable.
• Expression: The formula or function that determines the value of each term being added. This expression usually depends on the index variable.

General Form:

  ∑ Expression
 i=Lower Limit
      Upper Limit

This represents the sum of the "Expression" as the index variable 'i' ranges from "Lower Limit" to "Upper Limit" in increments of 1.

Examples:

• Arithmetic Series: ∑ i from i=1 to n, represents 1 + 2 + 3 + ... + n. This is a simple <a href="https://www.wikiwhat.page/kavramlar/arithmetic%20series">arithmetic series</a>.
• Geometric Series: ∑ r^i from i=0 to n, represents 1 + r + r^2 + ... + r^n. This is a <a href="https://www.wikiwhat.page/kavramlar/geometric%20series">geometric series</a>.

Properties:

• Linearity: ∑ (a*x_i + b*y_i) = a * ∑ x_i + b * ∑ y_i
• Constant Sum: ∑ c = n*c (where 'c' is a constant and the sum is from i=1 to n)

Applications:

Summation is used to calculate:

• <a href="https://www.wikiwhat.page/kavramlar/mean">Mean</a> and Variance in Statistics
• Area under a <a href="https://www.wikiwhat.page/kavramlar/curve">curve</a> (approximated by rectangles in <a href="https://www.wikiwhat.page/kavramlar/integral%20calculus">integral calculus</a>).
• Total work done by a force over a distance.
• Aggregating data in computer science algorithms.

Understanding summation notation is crucial for interpreting and manipulating mathematical formulas and algorithms across various scientific and technical disciplines.