Question

Evaluate each series. $$\sum_{k=1}^{6}(-1)^{k} \cdot k$$

   Evaluate each series.
$$\sum_{k=1}^{6}(-1)^{k} \cdot k$$

Precalculus

Margaret L. Lial,… 6th Edition

Chapter 11, Problem 43

Instant Answer

Solved by Expert Ankit Gupta

07/20/2020

Step 1

The series given is \(\sum_{k=1}^{6}(-1)^{k} \cdot k\). This means we need to evaluate the sum of the terms \((-1)^{k} \cdot k\) for \(k\) ranging from 1 to 6. Show more…

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Evaluate each series. $$\sum_{k=1}^{6}(-1)^{k} \cdot k$$

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Key Concepts

Exponentiation of -1

Exponentiating -1, as in (-1)^k, is a common method to introduce alternating signs into a series. The value of (-1)^k becomes -1 when k is odd and 1 when k is even, thus ensuring that successive terms in the series alternate in sign. This concept is foundational in understanding how alternating series are constructed and evaluated.

Series

A series is the sum of a sequence of terms. In mathematical notation, especially using the summation symbol (?), it represents adding up a finite (or sometimes infinite) list of numbers defined by a general term. Evaluating a series involves calculating the sum of these terms, often by identifying patterns or applying known formulas.

Summation Notation

Summation notation, expressed with the sigma symbol (?), is a compact way of representing the sum of a sequence of expressions. It includes an index of summation, an expression for the term being summed, and the lower and upper bounds of the index, which specifies which terms are included in the sum.

Alternating Series

An alternating series is a series in which the signs of the terms alternate between positive and negative. This is typically achieved by including a factor such as (-1)^k in each term, and it affects convergence properties and the approach for summing the series.

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