A series is the sum of the terms of a sequence. The sequence 3, 7, 11, 15 is a list; the series 3 + 7 + 11 + 15 = 36 is what you get by adding it. A finite series stops after n terms, written Sn. An infinite series has a sum only if its partial sums approach a fixed number.
In Spanish
serie
Also called
sum of a sequence, partial sum, arithmetic series, geometric series, infinite series
Pairing the first and last terms gives the arithmetic series formula. Filling a square shows why the infinite geometric series 12 + 14 + 18 + ⋯ adds up to exactly 1.
Formal definition and notation
For a sequence a1, a2, a3, …, the nth partial sum is Sn = a1 + a2 + ⋯ + an = Σnk=1ak. The infinite series Σ∞k=1ak converges to S if limn → ∞Sn = S; otherwise it diverges. An arithmetic series has Sn = n(a1 + an)2; a geometric series with ratior ≠ 1 has Sn = a1(1 − rn)1 − r, and for |r| < 1 the infinite sum is a11 − r.
A sequence is an ordered list, 1, 4, 9, 16. A series is the sum of its terms, 1 + 4 + 9 + 16 = 30.
Seriesvs.Sigma notation
Sigma notation is the shorthand for writing a series: Σ4k=1k2. The series is the sum it stands for.
Quick check: 3 questions
Question 1 of 3
What is 1 + 3 + 5 + ⋯ + 19?
Show answer and explanation
Answer: A. There are 10 odd numbers from 1 to 19, and S10 = 10(1 + 19)2 = 100. 81 counts only 9 terms; 190 multiplies 10 by 19; 110 adds an extra 10.
Question 2 of 3
What is the sum of the infinite series 8 + 4 + 2 + 1 + ⋯?
Show answer and explanation
Answer: C.r = 12, so S = 81 − 12 = 16. 15 is only the first four terms; the series does have a sum because |r| < 1.
Question 3 of 3
Which infinite series has a finite sum?
Show answer and explanation
Answer: D.9 + 3 + 1 + ⋯ has r = 13, so it converges (to 13.5). The others have ratios 2, 1 and 1.5, and their partial sums grow forever.
Series: frequently asked questions
What is the difference between a sequence and a series?
A sequence is an ordered list of numbers, separated by commas: 5, 10, 15, 20. A series is the sum of those numbers: 5 + 10 + 15 + 20 = 50. Every series comes from a sequence. Questions that ask for “the sum of the first n terms” are asking for a series.
How can an infinite series have a finite sum?
When the terms shrink fast enough, the running totals settle toward one number. In 12 + 14 + 18 + ⋯, the partial sums are 0.5, 0.75, 0.875, 0.9375, and they get as close to 1 as you like, so the sum is defined to be 1. A geometric series does this exactly when |r| < 1.
What is the formula for the sum of a geometric series?
For the first n terms, Sn = a1(1 − rn)1 − r, where a1 is the first term and r ≠ 1 is the common ratio. For an infinite geometric series with −1 < r < 1, the sum is S = a11 − r. Example: 6 + 2 + 23 + ⋯ = 61 − 13 = 9.
SequenceA sequence is an ordered list of numbers that follows a rule, such as 3, 7, 11, 15, …. Each number is a term, and its position is 1st, 2nd, 3rd and so on. You can describe a sequence with a formula for the nth term, an, or by how each term comes from the one before it.Functions · Grade 8, Algebra 1, Algebra 2RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8Sigma NotationSigma notation is a compact way to write a long sum using the Greek capital letter Σ (sigma). In Σ4k=1k2, the index k starts at the number below sigma, stops at the number above it, and each value is put into the rule k2: 1 + 4 + 9 + 16 = 30.Functions · Algebra 2, Precalculus, CalculusNumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2AverageAn average is one number that stands for the typical size of the values in a data set. In everyday math, average means the mean: add all the values, then divide by how many values there are. In statistics, the median and the mode are also kinds of average.Statistics · Grade 5, Grade 6, Grade 7DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3Common RatioThe common ratio is the fixed number you multiply each term of a geometric sequence by to get the next term. Find it by dividing any term by the term before it: r = anan − 1. In 5, 20, 80, 320, the common ratio is 4. It can be a fraction or negative, but never 0.Functions · Algebra 1, Algebra 2SumA sum is the result you get when you add two or more numbers. In 8 + 5 = 13, the numbers 8 and 5 are the addends and 13 is the sum. The order does not change it: 5 + 8 is also 13. Word problems ask for a sum with words like total, in all and altogether.Operations · Grade 1, Grade 2, Grade 3Arithmetic SequenceAn arithmetic sequence is a list of numbers in which you add the same amount each time to get the next term. That fixed amount is the common difference, d. In 4, 7, 10, 13, … the common difference is 3. The nth term is an = a1 + (n − 1)d.Functions · Grade 8, Algebra 1, Algebra 2Geometric SequenceA geometric sequence is a list of numbers where each term is the term before it times the same number, called the common ratio r. In 3, 6, 12, 24, … the ratio is 2. The nth term is an = a1 · rn − 1. The terms grow when |r| > 1 and shrink when |r| < 1.Functions · Algebra 1, Algebra 2, PrecalculusLimitA limit is the value a function gets closer and closer to as the input approaches a certain number. We write limx → af(x) = L. The function need not even be defined at a. For f(x) = x2 − 1x − 1, f(1) is undefined, but the limit as x → 1 is 2.Calculus · Precalculus, CalculusIntegralAn integral adds up infinitely many tiny pieces to find a total, such as the area under a curve. The definite integral ∫baf(x) dx is the signed area between the graph of f and the x-axis from x = a to x = b. An indefinite integral is an antiderivative: ∫ 2xdx = x2 + C.Calculus · CalculusCommon DifferenceThe common difference is the fixed amount added to each term of an arithmetic sequence to get the next term, written d. Find it by subtracting any term from the term after it: in 7, 11, 15, 19, d = 11 − 7 = 4. A positive d makes the terms increase, a negative d makes them decrease.Functions · Grade 8, Algebra 1, Algebra 2InfinityInfinity is the idea of something without any end or bound, written with the symbol ∞. It is not a real number, so you cannot add, subtract or divide with it like an ordinary number. Math uses it for endless sets, quantities that grow without limit, and intervals such as (3, ∞).Symbols and Notation · Grade 6, Algebra 1, Algebra 2