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Chapter 12: Problem 2

Evaluate. $$\sum_{k=1}^{4}(3 k-1)$$

Short Answer

Expert verified
The sum of the arithmetic sequence \(\sum_{k=1}^{4}(3 k-1)\) with terms 2, 5, 8, and 11 is \(26\).

Step by step solution

01

Identify the terms of the arithmetic sequence

Since the given arithmetic sequence is \( (3k-1) \) with k ranging from 1 to 4, we can find each term by substituting the values of k in the sequence: \( k = 1 : (3 \cdot 1 - 1) = 2 \) \( k = 2 : (3 \cdot 2 - 1) = 5 \) \( k = 3 : (3 \cdot 3 - 1) = 8 \) \( k = 4 : (3 \cdot 4 - 1) = 11 \) So the terms of the arithmetic sequence are 2, 5, 8, and 11.
02

Add the terms together to find the sum

Now that we have the terms of the arithmetic sequence, we can add them together to find the sum: \( \sum_{k=1}^{4}(3 k-1) = 2 + 5 + 8 + 11 \) = \( 26 \) The sum of the arithmetic sequence is 26.

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Key Concepts

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

Series Summation
In mathematics, a series summation is when you add up a set of numbers that follow a specific pattern or order. This is particularly useful when dealing with sequences, where each number is known as a term. Think of it like finding the total of a shopping list, where each item has a price, and you want to know the overall cost.

When you see a symbol like \( \sum \) in mathematics, it denotes summation, meaning you need to find the total sum of all the terms defined by the expression and limits provided.

  • The lower limit, usually denoted by something like \( k = 1 \), tells you where to start, and the upper limit tells you when to stop.
  • The expression next to \( \sum \) defines what each term looks like in the series.
By evaluating these terms, as shown in the exercise with \( \sum_{k=1}^{4}(3k-1) \), each value of \( k \) gives us a term of 2, 5, 8, and 11, which are summed to 26.
Arithmetic Series
An arithmetic series is a type of sequence where each consecutive term increases by a constant amount. This constant difference is known as the common difference. Imagine you are going up a staircase where each step is exactly the same height. In this way, you consistently add the same number to get to the next term.

To identify if a series is arithmetic, check if:
  • Each term is generated by adding a constant to the previous term.
  • The formula for each term follows a pattern, in this case, \( 3k - 1 \) is used with corresponding \( k \) values.
In the provided exercise, each term's difference is calculated by subtracting consecutive terms:

  • \(5 - 2 = 3\)
  • \(8 - 5 = 3\)
  • \(11 - 8 = 3\)
This common difference of 3 tells us the sequence is indeed arithmetic.
Step-by-step Evaluation
Breaking down a problem step-by-step is like following a simple recipe in a cookbook to make a dish. Each step needs to be handled carefully to ensure you reach the desired result smoothly.

  • **Identify the Terms**: Start by understanding how to get each term. For our problem, we substitute values of \( k \) from 1 to 4 into the expression \( 3k - 1 \).
  • **Calculate the Terms**: Perform the arithmetic to get each term. This resulted in terms like 2, 5, 8, and 11.
  • **Sum the Terms**: Add these terms together to get the total sum, which is 26 in our example.
This method of evaluation helps you avoid errors by checking each part of the process and ensuring you understand where each number comes from. It promotes clarity in solving sequences and series, like an arithmetic series.

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Most popular questions from this chapter

Use Taylor polynomials to estimate the following within 0.01. $$\sin 1$$

Sum the series. $$\sum_{k=0}^{\infty} \frac{1}{k !} x^{3 k+1}$$

(a) Use a graphing utility to draw the graph of the function $$f(x)=\left\\{\begin{aligned} e^{-1 / x^{2}}, & x \neq 0 \\ 0, & x=0 \end{aligned}\right.$$ (b) Use L' H貌pial's rule to show that for every positive integer \(n\) $$\lim _{x \rightarrow 0} \frac{e^{-1 / x^{2}}}{x^{n}}=0$$ (c) Prove by induction that \(f^{(n)}(0)=0\) for all \(n \geq 1\) (d) What is the Taylor series of \(f ?\) (e) For what values of \(x\) dei: \(s\) s the Taylor serics of \(f\) actually converge to \(f(x) ?\)

Estimate within 0.001 by series expansion and check your result by carrying out the integration directly. $$\int_{0}^{1 / 2} x \ln (1-x) d x$$

(a) Prove that if \(\sum_{k=0}^{\infty} a_{k}\) is a convergent series with all terms nonzero, then \(\sum_{x=0}^{\infty}\left(1 / a_{k}\right)\) diverges. (b) Suppose that \(a_{k}>0\) for all \(k\) and \(\sum_{k=1}^{\infty} a_{k}\) diverges. Show by example that \(\sum_{k=0}^{\infty}\left(1 / a_{k}\right)\) may converge and it may diverge.
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