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Browse High School Sequences, Series
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Selected answers to common questions:
Strategies for finding sequences.
- Second-Order Linear Recurrences [06/08/2001]
Three problems involving recurrence equations.
- Second Order Recurrence with Non-Constant Coefficients [05/27/2005]
I'm trying to find a closed form solution of a second order recurrence
relation with no constant coefficients, specifically: u(n+2) =
2*(2*n+3)^2 * u(n+1) - 4*(n+1)^2*(2*n+1)*(2*n+3)*u(n). Can you help?
- Sequence and Series Terminology and Concepts [11/27/2005]
I'm studying sequences and series and am confused about how they are defined in terms of functions as there seem to be some inconsistencies. Can you help clarify things for me?
- Sequence of Squares [07/25/1998]
Do you have any information on the sequence of squares?
- Sequence of Triangular Numbers [7/13/1995]
What is the sequence called 1, 3, 6, 10, 15 and how is it generated?
- Sequence Pattern and Closed Form [7/16/1996]
Given the pattern for a sequence, I can't figure out a general rule for
the nth term.
- Sequences [7/29/1996]
The sum of three numbers is 147 and when multiplied together they yield
21952... Find a formula for 60, 30, 20, 15...
- The Sequence Sin(n) [02/20/2002]
I am trying to prove that the sequence sin(n), for n, a natural number,
does not converge.
- Series Convergence [01/27/2001]
Test these series for convergence; if the series is alternating, tell
whether the convergence is conditional or absolute...
- Series Convergence [02/28/2001]
Why does 1 + 1/2^z + 1/3^z + ... converge for Re(z) greater than 1?
- Series Divergence [03/03/1999]
Show that the series sum(k=0 -> infinity): (k/e)^k/k! is divergent.
- Series Expansion of 1/(1-x) [08/01/1998]
Can you explain the series expansion identity 1/(1-x) = 1 + x + x^2 + x^3
+ ... ? In what region does it converge?
- Series for which Convergence is Unknown [11/09/2000]
Are there series for which it is unknown whether they converge or
- Series Problem: Find the Sum [6/24/1996]
Find sum[sin(nx)/(3^n),(n,0,oo)] if sin x=1/3 and x is in the first
- A Series that Converges and Diverges? [04/30/2002]
Let N = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + ... Does this series both
converge and diverge?
- Series Types [05/11/1997]
What are the definitions of convergent, divergent, and oscillating
- Showing That the Sum of the Infinite Series cos(n)/n Converges [04/01/2008]
I'm trying to determine if the sum of the series (cos n)/n for n = 1
to infinity converges. I've tried some tests but they have been
- Sigma Notation [4/14/1996]
I am trying to find questions regarding sigma notation.
- Sigma Notation [09/07/2001]
To prove that sigma (i^2) from i = 1 to n i equal to (n(n+1)(2n+1))/6
start with (i+1)^3 - i^3...
- A Simple Expression? [1/26/1996]
Is it possible to have a simple expression for a certain series starting
- Simple Number Pair Series Yields Surprising Ratio ... Why? [12/31/2009]
An enthusiast wonders about the curious ratio that emerges from a
simple pattern for generating number pairs. Doctor Rick builds an
algebraic argument for why its phi-like recursive relationship
approaches the square root of 2.
- Simplify a Geometric Series [05/06/2003]
x^n + x^(n - 1) + x^(n - 2) + ... + x^(n - n)
- Solutions to X^Y = Y^X [12/21/2000]
How can I find the solutions to the equation x^y = y^x? I have been told
that it involves the Lambert W Relation.
- Solving an Equation with Infinite Exponents [03/15/2007]
If x^x^x^x^x^x^x^x^x^x^x...... = 2, solve for x. How can I solve that
- Solving a Sequence [12/7/1995]
Write an expression to find the nth term of the following sequence: 3, 9, 18, 30, 45 . . .
- Solving Continued Fractions [08/08/1998]
How do you get sqrt(2) from 1/(2 + 1/(2 + 1/(2 + ...)))? How do you solve
continuous fractions in general?
- Some Algebra Problems [6/1/1996]
If z=(3-2i)^1/2 then find z^-...
- Square Root of 3 minus 1 [09/24/1997]
Express sqrt3-1 as a continued fraction.
- Square Root Theory [11/16/2001]
When I enter any positive number in the calculator or a fraction like
0.1, then take the square root of that number, then take the square root
of that number, and keep pressing the square root button over and over, I
eventually get to number 1. Why?
- Stair Patterns [02/27/2001]
The 1st step is made with 4 matches, the 2nd with 10 matches, the 3rd
with 18, the fourth with 28. How many matches would be needed to build 6,
10, and 50 steps?
- Subtracting Finite Sums of Integers [08/03/1998]
If n = 1 + 3 + 5 + 7 + ... + 999 and m = 2 + 4 + 6 + 8 + ... + 1000, what
does m-n equal?
- Summation by Parts [01/07/2004]
Using 'E' to represent sigma, is there an approximate solution to
E(Ai*Bi) = ? where i = 0,1,...,n if Ai is known explicitly and E(Bi)
- Summation Notation and Arithmetic Series [07/27/2001]
Do I need to use the arithmetic series formulas when doing sigma
- Summation of Floor Function Series [01/12/2009]
Is there a formula for the sum [p/q] + [2p/q] + [3p/q] + ... + [np/q]
where p, q, and n are natural numbers?
- Summation of Series: Faulhaber's Formula [07/30/2003]
I am asked to solve a series...
- Summations of n^(-2k) [09/10/2000]
How can I find the summations of the following series for n = 1 to
infinity: (n^-2), (n^-4), (n^-[2k]) and (n^-[2k+1])?
- Summing a Binary Function Sequence [07/16/1998]
How do you compute the sum of B(n)/(n(n+1)) from 1 to infinity, where
B(n) denotes the sum of the binary digits of n?
- Summing an Oscillating Series [08/10/1998]
Does 1 - 1 + 1 - 1 + 1 - ... equal 1 or 0
- Summing a Series Like n*(n!) [10/28/2001]
How can I add up a series like 1*1! + 2*2! + 3*3! ... n*n! ?
- Summing Consecutive Integers [08/30/1998]
Express 1994 as a sum of consecutive positive integers, and show that
this is the only way to do it.