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Chapter 14: Problem 36

Write out each series and evaluate it. $$ \sum_{i=1}^{5}(-1)^{i} \cdot i $$

Short Answer

Expert verified
-3

Step by step solution

01

Understand the series notation

The series \(\[\begin{equation} \sum_{i=1}^{5}(-1)^{i} \cdot i \end{equation}\]\) means we need to sum the values of \( (-1)^{i} \cdot i \) for each integer value of \( i \) from 1 to 5.
02

Compute each term in the series

Calculate each term individually: When \( i = 1 \), \( (-1)^1 \cdot 1 = -1 \) When \( i = 2 \), \( (-1)^2 \cdot 2 = 2 \) When \( i = 3 \), \( (-1)^3 \cdot 3 = -3 \) When \( i = 4 \), \( (-1)^4 \cdot 4 = 4 \) When \( i = 5 \), \( (-1)^5 \cdot 5 = -5 \)
03

Sum the computed terms

Add up the computed terms: \( -1 + 2 - 3 + 4 - 5 \). Group terms to simplify the addition: \( (-1 - 3 - 5) + (2 + 4) \). Calculate the sums: \( -9 + 6 = -3 \)
04

Write the final answer

The value of the series \(\[\begin{equation} \sum_{i=1}^{5}(-1)^{i} \cdot i \end{equation}\]\) is \(-3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Summation
Summation refers to the process of adding a sequence of numbers to get their total. For a given series, each term is computed individually and then all terms are summed together.
In our example, the series is \(\[\begin{equation} \sum_{i=1}^{5}(-1)^{i} \cdot i \end{equation}\]\). We first calculate each term:
  • When \(i=1\), \((-1)^{1} \cdot 1 = -1\)
  • When \(i=2\), \((-1)^{2} \cdot 2 = 2\)
  • When \(i=3\), \((-1)^{3} \cdot 3 = -3\)
  • When \(i=4\), \((-1)^{4} \cdot 4 = 4\)
  • When \(i=5\), \((-1)^{5} \cdot 5 = -5\)
Next, we sum these terms:
  • -1 + 2 - 3 + 4 - 5
Grouping the terms makes the calculation easier:
  • (-1 - 3 - 5) + (2 + 4) = -9 + 6
Which gives us:
  • -3
Hence, the value of the series is \(-3\).

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