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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.



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 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...

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?

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