summation

Examples of summation in the following topics:

• Series and Sigma Notation

  • Summation is the operation of adding a sequence of numbers, resulting in a sum or total.
  • For finite sequences of such elements, summation always produces a well-defined sum.
  • One way to compactly represent a series is with sigma notation, or summation notation, which looks like this:
  • In this formula, i represents the index of summation, $x_i$ is an indexed variable representing each successive term in the series, $m$ is the lower bound of summation, and $n$ is the upper bound of summation.
  • The "$i = m$" under the summation symbol means that the index $i$ starts out equal to $m$.

• Sums and Series

  • The summation of all the terms of a sequence is called a series, and many formulae are available for easily calculating large series.
  • Summation is the operation of adding a sequence of numbers; the result is their sum or total.
  • If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation.
  • where $i$ represents the index of summation; $x_i$ is an indexed variable representing each successive term in the series; $m$ is the lower bound of summation, and $n$ is the upper bound of summation.
  • The "$i=m$" under the summation symbol means that the index $i$ starts out equal to $m$.

• Applications and Problem-Solving

  • Using equations for arithmetic sequence summation can greatly facilitate the speed of problem solving.
  • To apply this to the first summation of all the even numbers between 50 and 100, we would want to add until the 50th term:

• Summing Terms in an Arithmetic Sequence

• Binomial Expansion and Factorial Notation

  • In order to solve this, we will need to expand the summation for all values of $k$.

• Binomial Expansions and Pascal's Triangle

  • Using summation notation, it can be written as:

• Partial Fractions

  • The denominators of the terms of this summation, $g_{j}(x)$, are polynomials that are factors of $g(x)$, and in general are of lower degree.