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Chapter 4: Problem 6

Write out all terms and compute the sums. $$\sum_{i=3}^{7}\left(i^{2}+i\right)$$

Short Answer

Expert verified
The sum of this series is 160.

Step by step solution

01

Understand the Series

The series we are asked to sum is given by the term \(i^{2}+i\). It's a finite series - the index \(i\) starts at 3 and goes up to 7. So, we have 5 terms in total that we need to calculate and sum.
02

Calculate Each Term

We now calculate each term in the series individually. \nFor \(i = 3\): \(3^2 + 3 = 9 + 3 = 12\)\nFor \(i = 4\): \(4^2 + 4 = 16 + 4 = 20\)\nFor \(i = 5\): \(5^2 + 5 = 25 + 5 = 30\)\nFor \(i = 6\): \(6^2 + 6 = 36 + 6 = 42\)\nFor \(i = 7\): \(7^2 + 7 = 49 + 7 = 56\)
03

Sum all the Terms

Finally, sum all the calculated terms: \(12 + 20 + 30 + 42 + 56 = 160\)

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Key Concepts

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

Finite Series
A finite series is a sum of a sequence of terms that has a definite beginning and a definite end. This means it doesn't go on forever, unlike an infinite series. In mathematical notation, a finite series is often expressed using the sigma notation \( \sum \). This notation is quite efficient because it specifies the expression to be summed, the starting point (lower limit), and the ending point (upper limit) of the index.

In our original exercise, the series \( \sum_{i=3}^{7}(i^2 + i) \) is finite. Here, the lower limit is 3, and the upper limit is 7, indicating that we'll calculate five terms in total.
Series Calculation
Calculating a series involves finding the values of each term in the series and then adding these values together. To do this smoothly, it is essential to follow a step-by-step approach. First, understand the individual terms in the expression. For example, in our series \( i^2 + i \), each term of the series depends on the current index value \( i \).

**Steps for Series Calculation:**
  • Identify the general term's expression, in this case, \( i^2 + i \).
  • Calculate each term separately for every value of \( i \) from the lower limit to the upper limit of the series.
  • Finally, sum these calculated values to obtain the total of the series.
This approach ensures no terms are missed and helps in systematically building up the final sum.
Index
The index is an essential component in a series as it represents the variable part of each term's expression, ensuring we calculate a series for different values. In mathematical expressions using the summation symbol \( \sum \), the index indicates the range over which calculations occur. The variable, usually represented by a letter like \( i \), \( n \), or \( k \), is adjusted at each step within its defined limits.

In the series \( \sum_{i=3}^{7}(i^2 + i) \), the index is \( i \), and it starts at 3 and increases by one until it reaches 7. This means each term is calculated by substituting the current index value into the expression \( i^2 + i \) before moving to the next index value. This mechanism ensures all specific instances within the series range are accounted for in the final calculation.

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Most popular questions from this chapter

Evaluate \(\int \frac{1}{|x| \sqrt{x^{2}-1}} d x\) by rewriting the integrand as \(\frac{1}{x^{2} \sqrt{1-1 / x^{2}}}\) and then making the substitution \(u=1 / x .\) Use your answer to derive an identity involving \(\sin ^{-1}(1 / x)\) and \(\sec ^{-1} x\)

Generalize exercises 56 and 57 to \(\int \frac{1}{x^{(p+1 / / q}+x^{p / q}} d x\) for positive integers \(p\) and \(q\)

The velocity of an object at various times is given. Use the data to estimate the distance traveled. $$\begin{array}{|l|r|r|r|r|r|r|r|} \hline t(\mathrm{s}) & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline v(t)(\mathrm{ft} / \mathrm{s}) & 40 & 42 & 40 & 44 & 48 & 50 & 46 \\ \hline \end{array}$$ $$\begin{array}{|l|r|r|r|r|r|r|} \hline t(\mathrm{s}) & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline v(t)(\mathrm{ft} / \mathrm{s}) & 46 & 42 & 44 & 40 & 42 & 42 \\ \hline \end{array}$$

Derive the formulas \(\int \sec ^{2} x d x=\tan x+c\) and \(\int \sec x \tan x \, d x=\sec x+c\)

In this exercise, we guide you through a different proof of \(\lim _{h \rightarrow 0} \frac{e^{h}-1}{h}=1 .\) Start with \(f(x)=\ln x\) and the fact that \(f^{\prime}(1)=1 .\) Using the alternative definition of derivative, we write this as \(f^{\prime}(1)=\lim _{x \rightarrow 1} \frac{\ln x-\ln 1}{x-1}=1 .\) Explain why this implies that \(\lim _{x \rightarrow 1} \frac{x-1}{\ln x}=1 .\) Finally, substitute \(x=e^{h}.\)
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