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Chapter 10: Problem 6

Write out each finite series. $$ \sum_{n=1}^{4}(-n)^{n} $$

Short Answer

Expert verified
The finite series sums to 232.

Step by step solution

01

Understand the Series Notation

The notation \( \sum_{n=1}^{4}(-n)^n \) represents a sum of terms where \( n \) varies from 1 to 4. In each term, \( n \) is raised to the power of \( n \) and then multiplied by \(-1\).
02

Calculate Each Term in the Series

Evaluate each term in the series for \( n = 1, 2, 3, 4 \).- For \( n = 1: \) \[ (-1)^1 = -1 \]- For \( n = 2: \) \[ (-2)^2 = 4 \] (since \((-2)^2 = 4\)).- For \( n = 3: \) \[ (-3)^3 = -27 \] (since \((-3)^3 = -27\)).- For \( n = 4: \) \[ (-4)^4 = 256 \] (since \((-4)^4 = 256\)).
03

Write Out the Finite Series

Using the results from Step 2, write out each term of the series: \( -1, 4, -27, 256 \).
04

Sum the Series

Add all the terms obtained in the previous steps to find the total sum:- First, sum the first two terms: \[ -1 + 4 = 3 \]- Then, add the third term: \[ 3 + (-27) = -24 \]- Finally, add the fourth term: \[ -24 + 256 = 232 \]
05

Confirm the Calculation

Double-check the calculation to ensure accuracy. The steps add up correctly, leading to the sum of all terms: \(-1 + 4 - 27 + 256 = 232\).

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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 specific number of terms. It comes to an end after a certain number of terms, unlike an infinite series that goes on forever. In the series notation \( \sum_{n=1}^{4}(-n)^n \), the number under the summation sign tells us where to start (\( n=1 \)), and the number on top tells us where to finish (\( n=4 \)).
Each term in this finite series is uniquely defined for every value of \( n \) from 1 to 4, making it a small and manageable sequence to sum up. Since it's finite, you can calculate the total by simply adding up all the terms. These series are helpful in understanding basic mathematical concepts before moving on to more complex infinite series.
Powers of Integers
When we talk about powers of integers, we are referring to a number raised to an exponent. In the given series \((-n)^n\), starting with \(-1\), means each integer \( n \) is raised to its own power which can significantly change the value.
  • For example, \((-1)^1\) results in \(-1\).
  • Considering \((-2)^2\), the calculation includes squaring \(-2\), and since a negative number squared becomes positive, the outcome is \(4\).
  • When \(n = 3\), we have \((-3)^3\), leading to a negative result of \(-27\) because a negative number cubed stays negative.
  • Finally, \((-4)^4\) raises \(-4\) to the fourth power, again resulting in a positive \(256\).
Calculating powers of integers is crucial since it forms the basis for many mathematical operations, and observing how the sign changes with different exponents is a crucial point when dealing with negative bases.
Summation
Summation is the process of adding a sequence of numbers. It's indicated by the Greek letter Sigma (\(\sum\)), and we use it to find the total of the finite series given. In our series \( \sum_{n=1}^{4}(-n)^n \), summation involves the following steps:
  • Calculate each term: \((-1)^1 = -1\), \((-2)^2 = 4\), \((-3)^3 = -27\), \((-4)^4 = 256\).
  • Write out and add all the terms: \(-1 + 4 - 27 + 256\).
  • The sum of these values is \(232\).
This illustrates how summation helps in collecting the result of a series of numbers into one total. It's a powerful concept used widely in mathematics for adding not just numbers, but functions, sequences, and more. Understanding summation is key for moving further into the topics of calculus and beyond.

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Most popular questions from this chapter

BUSINESS: Sinking Fund A sinking fund is an annuity designed to reach a given value at a given time in the future (often to pay off a debt or to buy new equipment). A company's \(\$ 100,000\) printing press is expected to last 8 years. If equal payments are to be made at the end of each quarter into an account paying \(8 \%\) compounded quarterly, find the size of the quarterly payments needed to yield \(\$ 100,000\) at the end of 8 years. [Hint: Solve \(\left.x+x(1.02)+x(1.02)^{2}+\cdots+x(1.02)^{31}=100,000 .\right]\)

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What is the maximum number of terms in the Taylor series for a polynomial of degree \(7 ?\)

ECONOMICS: Multiplier Effect If the marginal propensity to consume \((M P C)\) is \(70 \%\), then the marginal propensity to save (MPS) is \(30 \% .\) In general, \(M P S=1-M P C .\) Show that for a given \(M P C,\) the multiplier is \(\frac{1}{M P S}\)

GENERAL: Million Dollar Lottery Most state lottery jackpots are paid out over time (often 20 years), so the "real" cost to the state is the sum of the present values of the payments. Find the cost of a "million dollar" lottery by summing the present values of 240 monthly payments of \(\$ 4167,\) beginning now, if the interest rate is \(6 \%\) compounded monthly. [Hint: The present value of \(\$ 4167\) in \(k\) months is \(4167 /(1+0.005)^{k} .\)
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