Summary: Series and Their Notations

Key Equations

sum of the first [latex]n[/latex]
terms of an arithmetic series		 [latex]{S}_{n}=\frac{n\left({a}_{1}+{a}_{n}\right)}{2}[/latex]
sum of the first [latex]n[/latex]
terms of a geometric series		 [latex]{S}_{n}=\frac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}\cdot r\ne 1[/latex]
sum of an infinite geometric series with [latex]-1<r<\text{ }1[/latex] [latex]{S}_{n}=\frac{{a}_{1}}{1-r}\cdot r\ne 1[/latex]

Key Concepts

  • The sum of the terms in a sequence is called a series.
  • A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum.
  • The sum of the terms in an arithmetic sequence is called an arithmetic series.
  • The sum of the first [latex]n[/latex] terms of an arithmetic series can be found using a formula.
  • The sum of the terms in a geometric sequence is called a geometric series.
  • The sum of the first [latex]n[/latex] terms of a geometric series can be found using a formula.
  • The sum of an infinite series exists if the series is geometric with [latex]-1<r<1[/latex].
  • If the sum of an infinite series exists, it can be found using a formula.
  • An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series.

Glossary

annuity an investment in which the purchaser makes a sequence of periodic, equal payments

arithmetic series the sum of the terms in an arithmetic sequence

diverge a series is said to diverge if the sum is not a real number

geometric series the sum of the terms in a geometric sequence

index of summation in summation notation, the variable used in the explicit formula for the terms of a series and written below the sigma with the lower limit of summation

infinite series the sum of the terms in an infinite sequence

lower limit of summation the number used in the explicit formula to find the first term in a series

nth partial sum the sum of the first [latex]n[/latex] terms of a sequence

series the sum of the terms in a sequence

summation notation a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series

upper limit of summation the number used in the explicit formula to find the last term in a series