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Dr. Math FAQ:
Browse High School Sequences, Series
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Selected answers to common questions:
Strategies for finding sequences.
- Infinite Series Involving Pi [12/16/1997]
I need a reason why 1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + ..... = pi^2/6 is
correct. What if this series alternates?
- Infinite Series of 1/n [01/13/1998]
What is the sum of an infinite series of 1/n when n = 1,2,3...? I
understand the answer is divergence or the sum is infinity, but not why,
especially since the terms eventually go to 0.
- Infinite Series Problem [01/15/2002]
I have to find the correct answer to an infinite series question about
three people throwing dice...
- Infinite sums [6/26/1996]
Given the function f(x) = x^2/(1+x^2) find the sum f(1/n)+f(2/n)+...+
f(n-1/n)+f(n/n) for any positive integers "N".
- Integer Sequence [08/15/1997]
Show that if 19 distinct integers are chosen from the sequence
1,4,7,10,13,16,19...,97,100, there must be two whose sum is 104.
- Interesting Number Sequence Pattern [11/01/2004]
The sequence of digits 1,2,3,4,0,9,6,9,4,8,7,... is constructed in the
following way: every digit starting from the fifth is the last digit
of the sum of the previous four digits. (a) Do the digits 2,0,0,4
appear in the sequence in that order? (b) Do the initial digits
1,2,3,4 appear again in the sequence in that order?
- Interpolation and Extrapolation [08/10/1999]
Can you give me a step-by-step procedure for finding missing values based
on interpolation and extrapolation?
- Investigating Sequence Patterns [05/14/1998]
How can we find the pattern in the following sequence? Take a circle with
three dots on the circumference and connect with lines...
- Irrational Decimals [08/04/1998]
I know the proof stating that 0.9 (repeating) is actually equal to one,
but from a representation standpoint are they actually considered to be
- Isomorphisms [08/16/1999]
Can you give me an explanation and a nice example of isomorphism?
- Laurent Expansion [02/25/1999]
Explanation the Laurent Expansion using some examples.
- Laying a Brick Walkway [04/22/2002]
How many different ways can I build a walkway 2' by 20' of bricks 1'
by 2'? The bricks can lie vertically and horizontally, but in no other
- Laying Paving Stones [11/28/2001]
Finding a relation for a sequence that relates to the number of ways
paving stones can be laid to make a 3-foot-wide path using 3-foot by 1-
- Let f(x) = 1 + 1/2 + 1/3 + ... + 1/[(2^n)-1] [05/15/1999]
Which of the following inequalities are correct?
- Level of Medicine in the Human Body [11/1/1996]
A patient receives a 10mg dose of medicine every four hours... prove that
there will always be less than 40mg in the patient's body."
- The Limit of (1+1/x)^x As x Approaches Infinity [02/17/1998]
How Euler calculated e, and what it has to do with the equation
- Limit of (-1)^n? [03/14/1998]
As n approaches infinity, does (-1)^n have a limit?
- Limit of Area [03/01/1998]
Limit approached by area of a square when its sides are repeatedly
divided into three congruent parts and squares are constructed outwardly
on the middle parts.
- Limit of a Series [03/02/2003]
What is the limit of the following series: a(n) = ((1^n) + (2^n) + ... +
- Limits - Indeterminate Forms [10/12/1997]
I cannot do a problem where I need to convert into the form 0/0 and then
use L'Hopital's Rule...
- Limits of Sequences [02/25/2001]
Is the limit of [(1 + 1/sqrt(n))^(1.5n)], as n goes to infinity, e? What
is the limit as n goes to infinity of [(1 + a/n)^n], where a is not equal
- Limits of Sequences [08/19/1997]
Please explain the limit superior of a sequence .
- Linear Recurrance Relations [08/10/2004]
Is there a general approach to taking a pattern that is defined
recursively and finding an explicit definition for it?
- Looking for Patterns [10/30/2001]
What would be the answer to: (x-a)(x-b)(x-c)(x-d)(x-e)...(x-y)(x-z)?
- Maclaurin Series for Tangent [09/17/2001]
What is the Maclaurin series for tangent (not inverse tangent)?
- Making a Series Sum to Zero [05/24/2002]
How can I place + and - signs between 1^2, 2^2, 3^2, ..., 2005^2 to
make the sum equal zero?
- Mangoes at the Gates [04/06/2001]
To pick some mangoes from a tree inside seven walls with seven guards,
you tell each guard that you'll give him half of the mangos you have, but
he must give you back one mango. What's the minimum number of mangos you
must pick to have at least one mango left?
- Mathematical Series [04/19/2005]
Given the series mtbf = 1/a + 1/2a + 1/3a + .... + 1/na with mtbf =
23998 and a = 0.0008, how do I solve for n?
- Math Virus Formula [10/23/2001]
The virus spreads to all the squares directly touching each other (not
including diagonally) and I have found the formula for the number of
newly infected cells (although this does not include the first minute)...
- Mean of a Sequence [06/09/2002]
What is the arithmetic mean of the first one thousand positive odd
- Method of Finite Differences [10/12/2000]
How can I find the generating equation for the series -3, 2, 13, 30, 53?
- The Method of Finite Differences, Extended -- and Shortened [12/22/2010]
Given the first four terms of two series, a student struggles to generate polynomials that describe them with the method of finite differences. Doctor Greenie shows her how to create a row with a common difference -- and save effort with a shortcut.
- Millionth Digit of the Counting Numbers [02/26/2001]
A number is formed by writing the counting numbers in order:
123456789101112131415... What is the one millionth digit in this number?
- Millionth Term [04/16/2001]
What is the millionth term in the sequence 1, 2, 2, 3, 3, 3, ... ?
- Millionth Term of a Sequence [06/18/2002]
what would be the millionth term in the sequence 1, 2, 2, 3, 3, 3, 4,
4, 4, 4,...?
- A Monster of a Continued Fraction [11/09/1996]
How do you find the value of a continued fraction?
- Multiplying Mice [07/23/1997]
Baby mice can breed when they are 6 weeks old and the babies are born
after 3 weeks. If each mother mouse has only one litter and all the
litters have 8 babies, half males and half females, how many mice will
you have 18 weeks from today?
- Naming Geometric and Arithmetic Progressions [04/04/2003]
Why is an exponential progression called 'geometric'? Why is a linear
progression called 'arithmetic'?
- Natural Numbers [01/08/1998]
What are two ways of finding the sum of n natural numbers?
- Nested Radical [02/25/2002]
Prove the following nested radical:
sqrt(1+2sqrt(1+3sqrt(1+4sqrt(1+...)))) = 3