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Chapter 11: Problem 48

Use a graphing calculator to evaluate each series. $$\sum_{i=1}^{10}\left(i^{3}-6\right)$$

Short Answer

Expert verified
\(\sum_{i=1}^{10}(i^{3} - 6) = 920\)

Step by step solution

01

- Understand the Series

The series to be evaluated is the sum of the expression \(i^{3} - 6\) as \(i\) goes from 1 to 10.
02

- Input the Series into the Graphing Calculator

Use a graphing calculator's summation function. Most graphing calculators have a \(\sum\) function that allows for summation of expressions. Input \(\sum_{i=1}^{10} (i^{3} - 6)\).
03

- Calculate Each Term Individually (Optional)

If a graphing calculator is not available, manually compute each term: \(1^{3} - 6, 2^{3} -6, \ldots, 10^{3} - 6\), and then sum these results.
04

- Evaluate the Series

Using the calculator, find the sum of the series. The expression should yield the sum for all terms from \(i = 1\) to \(i = 10\).

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Key Concepts

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

summation notation
Summation notation is a mathematical symbol used to represent the sum of a series of terms. It is often denoted by the Greek letter Sigma ( \sum ). The general format is \sum_{i=a}^{b} f(i), which means that you sum the function f(i) starting from the lower limit a up to the upper limit b. For example, if you see \sum_{i=1}^{10} (i^{3} - 6), you should interpret it as adding up all the terms of the expression \(i^{3} - 6\) from i=1 to i=10. Summation notation is useful as it provides a compact way to represent the sum of multiple terms without writing out each term individually. This can make working with series both more efficient and less error-prone.
graphing calculator functions
Graphing calculators are powerful tools that can simplify complex calculations, including evaluating series. Most graphing calculators have built-in functions to handle summation. To evaluate a series like \sum_{i=1}^{10} (i^{3} - 6), you need to locate the summation function, often marked as 危 or SUM on the calculator. Here鈥檚 the procedure:
  • First, access the summation function.
  • Then, enter the expression you want to sum, in this case, \(i^{3} - 6\).
  • Set the limits for i, starting at 1 and ending at 10.
  • Finally, execute the function to get the result.
Graphing calculators not only perform these calculations quickly but also reduce the risk of manual errors. Additionally, they can store and recall expressions, making repeated calculations more efficient.
step-by-step problem solving
When faced with a mathematical problem like evaluating a series, breaking it down into smaller steps can make the process clearer and more manageable. Let's illustrate this with your given series \sum_{i=1}^{10} (i^{3} - 6):
  • Step 1: Understand the Series: Recognize that the series involves summing the expression \(i^{3} - 6\) from i=1 to i=10.

  • Step 2: Input the Series into the Graphing Calculator: Use the summation function on your graphing calculator. Enter \sum_{i=1}^{10} (i^{3} - 6) as instructed by your device's manual.

  • Step 3: Calculate Each Term Individually (Optional): Although using a calculator is convenient, understanding the underlying calculations is valuable. You could manually compute each term from i=1 to i=10 and sum them up: \(1^{3} - 6, 2^{3} - 6, \ldots, 10^{3} - 6\).

  • Step 4: Evaluate the Series: Finally, use the calculator to find the sum of the series from i=1 to i=10. The use of technology speeds up the process and helps ensure accuracy.
Following these steps can make complex problems simpler and improve your problem-solving skills overall.

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Most popular questions from this chapter

Use the summation properties and rules to evaluate each series. $$\sum_{i=1}^{4}\left(3 i^{3}+2 i-4\right)$$

You are offered a 6 -week summer job and are asked to select one of the following salary options. Option I: 5000 dollars for the first day with a 10,000 dollars raise each day for the remaining 29 days (that is, 15,000 dollars for day 2,25,000 dollars for day \(3,\) and so on) Option 2: 0.01 dollars for the first day with the pay doubled each day (that is, 0.02 dollars for day \(2,\) 0.04 dollars for day 3, and so on. Which option would you choose?

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