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Chapter 11: Problem 39

Evaluate each series. $$\sum_{i=-2}^{3} 2(3)^{i}$$

Short Answer

Expert verified
The series sums to 81.

Step by step solution

01

Write Down the Series

The given series is \(\sum_{i=-2}^{3} 2(3)^i\). This notation represents the sum of the expressions \2(3)^i\ from \i = -2\ to \i = 3\.
02

Evaluate Each Term Individually

Calculate each term from \i = -2\ to \i = 3\: \(2(3)^{-2} = 2 \cdot \frac{1}{9} = \frac{2}{9}\) \(2(3)^{-1} = 2 \cdot \frac{1}{3} = \frac{2}{3}\) \(2(3)^{0} = 2 \cdot 1 = 2\) \(2(3)^{1} = 2 \cdot 3 = 6\) \(2(3)^{2} = 2 \cdot 9 = 18\) \(2(3)^{3} = 2 \cdot 27 = 54\)
03

Add All Calculated Terms Together

Add up the results from each term: \(\frac{2}{9} + \frac{2}{3} + 2 + 6 + 18 + 54\) To combine these, convert fractions to a common denominator and add:\(\frac{2}{9} + \frac{6}{9} + 2 + 6 + 18 + 54\)\(\approx 0.222 + 0.667 + 2 + 6 + 18 + 54\)
04

Simplify the Sum

Combine all terms to find the total sum: \(\approx 0.222 + 0.667 + 2 + 6 + 18 + 54 = 80.999\), which rounds to 81.

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Key Concepts

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

Geometric Series
A geometric series is a series of terms where each term is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In this exercise, the geometric series involves the terms generated by the function \(2(3)^i\). Here, the common ratio is 3, and the first term when \(i = -2\) is \(2(3)^{-2}\). Remember:
  • If the common ratio \(r\) is between -1 and 1, the series converges (sums to a finite number).
  • If \(r\) is greater than 1 or less than -1, the series diverges (sums to infinity).

So our task involves calculating terms from \(i = -2\) to \(i = 3\) and then summing them.
Summation Notation
Summation notation, represented by the Greek letter sigma (\(\Sigma\)), is a concise way to represent the sum of a sequence of numbers. In our exercise, the given summation notation \(\sum_{i=-2}^{3} 2(3)^i\) tells us:
  • \(i\): The index of summation, starting from \(-2\) up to \(3\).
  • \(2(3)^i\): The general term of the sequence.
To evaluate this, calculate each term individually then add them. Summation notation is particularly useful in defining sequences mathematically and can simplify complex series evaluations. Always check your upper and lower limits carefully.
Exponential Functions
Exponential functions are mathematical functions of the form \(a^x\), where \(a\) is a constant base and \(x\) is the exponent. In this exercise, our base is 3, and the exponent \(i\) ranges from -2 to 3. Here are key properties of exponential functions:
  • For positive bases: A positive exponent increases the value.
  • For negative exponents: The function value is the reciprocal (e.g., \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\)).
Exponential functions grow rapidly with positive exponents and decay rapidly with negative ones. So, we observe a combination of growth and decay in our series.

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Most popular questions from this chapter

Assume that \(n\) is a positive integer. Use mathematical induction to prove each statement S by following these steps. See Example \(I\). (a) Verify the statement for \(n=1\) (b) Write the statement for \(n=k\) (c) Write the statement for \(n=k+1\) (d) Assume the statement is true for \(n=k\). Use algebra to change the statement in part (b) to the statement in part (c). (e) Write a conclusion based on Steps (a)-(d). $$3+9+27+\dots+3^{n}=\frac{1}{2}\left(3^{n+1}-3\right)$$

Use the sequence feature of a graphing calculator to graph the first ten terms of each sequence as defined. Use the graph to make a conjecture as to whether the sequence converges or diverges. If you think it converges, determine the number to which it converges. $$a_{n}=(1+n)^{1 / n}$$

Assume that \(n\) is a positive integer. Use mathematical induction to prove each statement S by following these steps. See Example \(I\). (a) Verify the statement for \(n=1\) (b) Write the statement for \(n=k\) (c) Write the statement for \(n=k+1\) (d) Assume the statement is true for \(n=k\). Use algebra to change the statement in part (b) to the statement in part (c). (e) Write a conclusion based on Steps (a)-(d). $$5 \cdot 6+5 \cdot 6^{2}+5 \cdot 6^{3}+\dots+5 \cdot 6^{n}=6\left(6^{n}-1\right)$$

Alicia drops a ball from a height of \(10 \mathrm{m}\) and notices that on each bounce the ball returns to about \(\frac{3}{4}\) of its previous height. About how far will the ball travel before it comes to rest? (Hint: Consider the sum of two sequences.)

Work each problem.One game in a state lottery requires you to pick 1 heart, 1 club, 1 diamond, and 1 spade, in that order, from the 13 cards in each suit. What is the probability of getting all four picks correct and winning \(\$ 5000 ?\)
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