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 TOPICS This page:   sequences/series    Search   Dr. Math See also the Dr. Math FAQ:   golden ratio,   Fibonacci sequence Internet Library:   sequences/sets HIGH SCHOOL About Math Analysis Algebra    basic algebra    equations/graphs/      translations    linear algebra    linear equations    polynomials Calculus Complex Numbers Calculators/    Computers Definitions Discrete Math    permutations/    combinations Exponents    Logarithms Fibonacci Sequence/   Golden Ratio Fractals Functions Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean    practical geometry    symmetry/tessellations History/Biography Interest Logic Negative Numbers Number Theory Physics/Chemistry Probability Projects Puzzles Sequences/Series Sets Square/Cube Roots Statistics Transcendental   Numbers Trigonometry Browse High School Sequences, Series Stars indicate particularly interesting answers or good places to begin browsing. Selected answers to common questions:     Strategies for finding sequences. Taylor Series Expansion [11/24/2001] A distance from A to B is 1000 meters. As one traverses it at 1 meter per second, the distance is instantaneously and uniformly stretched 1000 additional meters. How long does it take to get from A to B? Telescoping Series [12/30/1996] Find the sum (to the nth term) of: 1/(1x3) + 1/(3x5) + 1/(5x7) +....+ 1/{(2n-1)(2n+1)} Telescoping Series [03/28/2001] Find the nth term in this pattern... Terms of the Series 1/n [05/03/2001] How many terms of the series 1/1 + 1/2 + 1/3 + ... + 1/n would I need to guarantee that the sum will be larger than some given value x? Test for Convergence [8/20/1996] Sum {from k=0 to infinity} [{log(k+1)-log k}/tan^(-1) (2/k)] Triangle Series [07/09/1999] What is the formula for finding the sum of the nth row of a triangle of numbers? Triangular Numbers: Find n As a Function of s [10/13/2002] s = 2, 3, 4, 5; n = 1, 4, 10, 20. I need to find the relation between s and n, where n is the subject, i.e. n = .... Triangular Numbers in a Proof [04/08/1997] How do you prove 1^3+2^3+3^3+ ...+n^3 = (1+2+3+...+n)^2 by induction? Tribonacci Numbers [11/11/2000] Is there an implicit formula to calculate the nth Tribonacci number? Also, is there a formula to find the sum of the first n Tribonacci numbers? Tricks to Sum an Infinite Series [01/23/1999] What does the sum of [(-1)^n*|sin n|]/n from n = 1 to infinity converge to? Tricky Infinite Series and Radius of Convergence [04/21/2008] We've found a radius of convergence for the sum of the infinite series (4/3)^n * [n!(1*4*7*10...{3n-2})] / [(2n)!], but have not been able to analyze the right endpoint of that radius. Can you help? Trigonometric Equation for a Sequence [04/03/2001] I need an equation for the sequence 0, 0, 1, 0, 0, 1, 0, 0, 1, ... using the sine or cosine function. The Troublesome Endpoints of a Trigonometric Series [04/10/2011] A student has doubts about the radius of convergence assumed in a proof that involves integrating a power series. Doctor Vogler confirms his suspicion about the behavior of endpoints of integrated series, and introduces Abel's Theorem. Two Arithmetic Series [7/1/1996] an and bn are two arithmetic series; An and Bn are sums of first n elements. If (4n+27) * An = (7n+1) * Bn, compute a11/b11. Two Ways to Find a Formula [04/14/1998] I need to show that Sigma(rx^r) = (x-(n+1)x^(n+1)+nx^(n+2))/(1-x)^2. Unique Subset of Set of Fractions [7/19/1996] How can I determine a set of fractions such that if I add any subset of those fractions, I get a result that is unique relative to the result of any other subset in this set? Using Sine and Tangent to Find Pi [08/01/1998] How can I determine the sine or tangent of an angle without using a calculator? Using the Comparison Test to Determine if a Series Converges [05/19/2005] A discussion of how to apply the Comparison Test in general and in specific to the series sum n = 3 to infinity of 1/[[ln(n)]^ln(n)]. Using the Geometric Mean in a Sequence [07/11/2000] How do you use the geometric mean to find the missing number in the geometric sequence 5,15,_,135,405? Volume of Water in an Urn [11/12/1996] An urn contains 1 liter of water; a second urn nearby is empty. After pouring the water back and forth 1,200,000 times in a certain way, how much water is left in each urn? What are Taylor series? [04/18/1999] What are Taylor Series? What are they needed for? What Comes Next? Finite Differences [08/20/2015] A job candidate wants to become better at solving puzzles that ask for the next number in a sequence. Emphasizing how many valid rules might exist to generate justifiable solutions, Doctor Greenie introduces an elementary method based on repetitive subtraction. What Comes Ninth? [11/07/2010] A student wants to know the next number in a sequence of integers. Doctor Ali reveals a pattern. What is a continued fraction? [03/06/1998] What is a continued fraction and what makes it different from the types of fractions or ratios I'm used to? What is Meant by "the Sum of a Series"? [10/21/2003] A series is the sum of consecutive terms. So why do many authors speak about "the sum of a series" if the name "series" already means "sum of consecutive terms"? What is the nth Term? [08/02/1997] The first four terms of a sequence are 16, 8, 4, 2. Find the next two terms and the rule for the nth term. Which Elements are Divisible by 11? [07/12/1999] Determine all solutions w = x+y+z, (1/w) = (1/x)+(1/y)+(1/z). In a given sequence, find which elements of the sequence are divisible by 11. Why is a Telescoping Series Called Telescoping? [03/29/2008] I understand what a telescoping series is, but I'm curious why it is referred to as 'telescoping'. What's the reason for that name? Why Study Sequences and Series? [02/26/1999] What are some applications of arithmetic sequences and series? Why Use Radians instead of Degrees? [09/25/2001] Consider Maclaurin's theorem, from which we derive the Taylor series... Working with Sequences [11/14/1998] Give the next two terms of the sequence: 1, 1, 2, 4, 3, 9, .... Writing Sigma Notation [02/03/1999] What does the equation that goes under the Sigma notation imply? Wrong Number in a Sequence [08/13/1999] In the sequence {1, 1, 2, 2, 6, 18, 21, 84, 88, ...} which number is incorrect? What number does it need to be replaced with? Page: [

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