Question

Express the sums in closed form. $$\sum_{k=1}^{n-1} k^{2}$$

   Express the sums in closed form.
$$\sum_{k=1}^{n-1} k^{2}$$

Calculus A New Horizon

Calculus A New Horizon

Howard Anton 6th Edition

Chapter 7, Problem 24

Instant Answer

Solved by Expert Manisha Sarker

07/02/2021

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We know that the formula for the sum of squares of the first n natural numbers is given by $\frac{n(n+1)(2n+1)}{6}$. Show more…

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Express the sums in closed form. $$\sum_{k=1}^{n-1} k^{2}$$

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Key Concepts

Polynomial Summation Formulas

Polynomial summation formulas are derived expressions that give the sum of powers of integers for a specific exponent. For example, formulas for the sum of the first n natural numbers, their squares, or cubes are classic examples, and these formulas play a critical role in number theory and combinatorics.

Sum of Squares Formula

The sum of squares formula is a specific closed form that calculates the sum of the squares of the first n integers or a subset of them. Such formulas are instrumental in solving problems related to series, sequence analysis, and in the evaluation of quadratic expressions in discrete mathematics.

Summation Notation

Summation notation is a compact way to represent the addition of a sequence of numbers, using the sigma symbol (?) to denote the sum of terms produced by a function over a specified range of indices. This notation is essential for expressing aggregated sums in a clear and concise mathematical format.

Closed Form Expression

A closed form expression is an explicit equation or formula that computes the value of a sum or sequence without the need for iterative summation. It typically involves a finite number of standard operations and functions, and it simplifies the evaluation and analysis of sums.

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