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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 Expansion [11/21/2001]
Can you give me the proof of this statement: arcsin(x) = x + 1/2 (x^3/ 3) + (1/2)(3/4)(x^5/5) + (1/2)(3/4)(5/6)(x^7/7) + ...? The basis of the calculation is a Taylor series.

Taylor series [11/3/1994]
Please describe the Taylor series.

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?

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