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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/{(2n1)(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...{3n2})] / [(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))/(1x)^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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