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Chapter 11: Problem 5

Find the sum of each series. \(\sum_{i=1}^{5} i^{2}\)

Short Answer

Expert verified
The sum of the series is 55.

Step by step solution

01

Understand the Series Notation

The series notation \(\sum_{i=1}^{5} i^{2}\) represents the sum of the squares of the first 5 positive integers. This means we need to find the sum of \(1^{2}, 2^{2}, 3^{2}, 4^{2}, 5^{2}\).
02

Calculate Each Term

Calculate the square of each integer from 1 to 5: \(\begin{align*}1^{2} &= 1 \2^{2} &= 4 \3^{2} &= 9 \4^{2} &= 16 \5^{2} &= 25\end{align*}\)
03

Add the Calculated Terms

Add the results from Step 2 together: \(1 + 4 + 9 + 16 + 25\).
04

Compute the Final Sum

Perform the addition: \(1 + 4 = 5, \ 5 + 9 = 14, \ 14 + 16 = 30, \30 + 25 = 55\). Hence, the sum of the series is 55.

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Key Concepts

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

series notation
Series notation is a concise way to represent the sum of a sequence of terms. It uses the Greek capital letter Sigma \(\textstyle\sum\) to indicate the summing process. For example, \(\sum_{i=1}^{5} i^{2} \) means we are summing the square of integers from 1 to 5.
This way of representing sums helps us easily understand and communicate large or complex sums.
Here, the letter 'i' is called the index of summation, and it starts at 1 (the lower limit) and goes up to 5 (the upper limit). The term at each index is squared in this series. Together, the notation tells us quickly and clearly what to add up.
square of integers
The square of an integer is the integer multiplied by itself. For example, the square of 2 is 2 * 2 = 4.
In general, the square of an integer 'n' is represented as \(n^{2}\). Squaring numbers is an important part of many mathematical operations and appears frequently in series notation.
For the series \(\sum_{i=1}^{5} i^{2}\), we calculate:
  • 1^2 = 1
  • 2^2 = 4
  • 3^2 = 9
  • 4^2 = 16
  • 5^2 = 25
This gives us the individual terms that will be summed together.
summation of series
Summation of series involves adding up all the terms in the sequence indicated by the series notation.
For \(\sum_{i=1}^{5} i^{2}\), we calculate the sum of the squares of each integer from 1 to 5. Our terms are:
1, 4, 9, 16, and 25.
To find the sum, we simply add these together step by step. We perform the arithmetic:
  • 1 + 4 = 5
  • 5 + 9 = 14
  • 14 + 16 = 30
  • 30 + 25 = 55
Thus, the summation of the series gives us a total sum of 55.
arithmetic operations
Arithmetic operations are the basic computations of addition, subtraction, multiplication, and division.
In calculating the sum of the series \(\sum_{i=1}^{5} i^{2}\), we mainly use addition. We first square each integer and then add the results together one by one.
Breaking the addition into smaller steps can make it more manageable:
  • 1 + 4 = 5
  • 5 + 9 = 14
  • 14 + 16 = 30
  • 30 + 25 = 55
each of these steps uses simple addition, which is one of the fundamental arithmetic operations.
We end up with the final sum, which is 55.

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