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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.

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) is known?

Summation Notation and Arithmetic Series [07/27/2001]
Do I need to use the arithmetic series formulas when doing sigma notation?

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.

Summing Exponents [06/11/2002]
If you multiply successive powers of 3 (i.e., 3^1 * 3^2 * ...) to get 3^210, what is the final power of 3 in the list?

Summing Integers to the Fourth Power [09/26/1998]
How do you find the formula for the sum of integers to the fourth power: 1^4 + 2^4 + ... +n^4?

Summing n^k [11/24/1998]
Is there a general formula for summing the n^k, where k is a positive integer?

Summing Odd Numbers Geometrically [10/30/1999]
Can you prove that 1 + 3 + 5 + ... + (2n-1) = n*n by using a simple geometric method?

Summing the Integers from 1 to n [08/31/2005]
How do you add up the integers from 1 to n by using a formula instead of adding them all up the long way?

Summing Triangle Numbers [04/21/1998]
Can you help me find the formula to find the sum of a finite number of triangle numbers?

Sum of 1/n^2 [07/24/2000]
Computing the sum of 1/n^2 without using Fourier series.

Sum of 1/Sqrt(i) [11/20/2000]
What is the formula for the sum of 1/sqrt(i) for i = 1 to n? Can you show me the proof by induction?

Sum of a Geometric Progression [12/01/2002]
How can an infinite sum of positive numbers be negative?

Sum of An Infinite Series [07/08/1998]
Is it possible to add up all the terms of an infinite series?

Sum of Any Infinite Series [11/19/2000]
Even if you can determine that a series converges, it's usually impossible to calculate the sum exactly. Why?

Sum of a Power Series [02/10/2001]
How can I calculate the sum of the power series x + 4x^2 + 9x^3 + 16x^4 + ... + n^2x^n + ...?

Sum of a Sequence [06/10/1999]
How would we find the sum of the sequence (3,4,6,9,13, ..., 499503)?

Sum of a Series [10/26/1996]
Compute the sum of the coefficients of the expansion of (x+0.5)^100 for which the exponent is divisible by three.

Sum of Consecutive Cubes [05/11/2000]
How can I prove that the sum of consecutive cubes equals a square? That is, 1^3 + 2^3 + 3^3 + ... + n^3 = m^2.

Sum of Consecutive Odd Integers [07/27/2001]
Given an integer N, can N can be written as a sum of consecutive odd integers? If so, how can I identify *all* the sets of consecutive odd integers that add up to N?

Sum of Consecutive Squares [05/11/2001]
The sum for i = 1 to n, of i^2, is equal to ((n)(n+1)(2n+1))/6. Why?

Sum of Convergent Series [09/25/1999]
How can you find the sum for k = 0 to infinity of 1/[(k+1)(k+3)], and the sum for k = 0 to infinity of [(25/10^k) - (6/100^k)]?

Sum of Fibonacci Series [05/23/2000]
Is there a formula for the sum of the first n numbers in the Fibonacci sequence?

Sum of First n Odd Numbers [7/10/1996]
Show that the sum of the first n odd numbers is a perfect square.

Sum of Harmonic Series [5/9/1996]
What's the total sum in terms of variables for the series (1/1*2)+(2/2*3)......+n/n(n+1)?

Sum of i [03/23/2002]
If the sum as i goes from 1 to n of 2^i is 2^n -1, what is the sum as i goes from 1 to n of 3^i ?

Sum of Inverse of Primes [05/25/1999]
Is the infinite series S = 1/1 + 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + ... + 1/p(n) + ... convergent or divergent?

Sum of Sine Series [07/17/2008]
Show that for any integer n >= 1, the sum from k = 1 to n of sin(kt) is [cos(1/2t) - cos((n + 1/2)t)]/[2 sin(1/2t)].

Sum of Some Doubling [08/18/2015]
How do you determine the sum of the counting numbers 1, 2, 3, ...n, where each term appears 2n times? Doctor Ali shows the way forward, beginning with the formula for the first n integers.

Sum of Squares Derivation [11/30/2002]
I am looking for a derivation of the formula for the sum of the first n squares.

Sums of Consecutive Integers [02/04/2001]
What numbers can be expressed as the sum of a string of consecutive positive integers?

Sums of Consecutive Positive Integers [03/02/2001]
Why are the powers of 2 the only numbers you cannot get as the sum of a series of consecutive positive integers?

Surveying Sum Strategies [02/26/2011]
A student who knows how to solve some sums seeks a method general enough to determine the explicit generating formula for any finite sum. Doctor Vogler confirms the student's hunch that no single approach suffices before linking to a selection of other Dr. Math conversations that address most kinds of finite sums.

Taylor approximation of tan^2(x) [6/12/1996]
Just to check that I can't do this because f'(0) = infinity...

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