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Chapter 11: Problem 82

Calculate each of the following sums. $$\sum_{k=1}^{\infty} 5(-0.7)^{k}$$

Short Answer

Expert verified
-2.06

Step by step solution

01

Identify the type of series

This is an infinite geometric series with the general form \(\sum_{k=1}^{\infty} ar^k\), where \(a\) is the first term and \(r\) is the common ratio.
02

Recognize the first term and the common ratio

From the given series \(\sum_{k=1}^{\infty} 5(-0.7)^k\), we can identify that \(a = 5(-0.7)^1 = -3.5\) (since it's the first term) and \(r = -0.7\).
03

Check if the series converges

An infinite geometric series converges if the absolute value of the common ratio is less than 1, i.e., \(|r| < 1|\). Here, \(|-0.7| = 0.7 < 1\), so the series converges.
04

Use the formula for the sum of an infinite geometric series

The sum of an infinite geometric series \(\sum_{k=1}^{\infty} ar^k\) is given by \(\frac{ar}{1-r}\).
05

Calculate the sum

Substitute \(a = 5\) and \(r = -0.7\) into the formula: \[\sum_{k=1}^{\infty} 5(-0.7)^k = \frac{5(-0.7)}{1-(-0.7)} = \frac{-3.5}{1+0.7} = \frac{-3.5}{1.7} = -2.0588\approx -2.06.\]

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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 sum of terms where each term is found by multiplying the previous term by a constant called the common ratio, denoted by \( r \). For example, in the series \( 1 + 2 + 4 + 8 + 16 + ... \), each term is twice the previous term, so the common ratio is 2. The general form of a geometric series is \(\textstyle \sum_{k=0}^{\text{n}} ar^k \) where \( a \) represents the first term and \( r \) is the common ratio. If the series continues infinitely, like in the provided exercise, it becomes an infinite geometric series.
series convergence
For an infinite geometric series to converge (meaning it sums up to a finite value), the absolute value of the common ratio must be less than 1, i.e., \(| r | < 1 \). In simpler terms, the terms in the series must get smaller and smaller in magnitude. For instance, in the given exercise, the series is \(\textstyle \sum_{k=1}^{\infty} 5(-0.7)^k \). Here, \( r = -0.7 \) and \(| -0.7 | = 0.7 \), which is indeed less than 1. Thus, the series converges.
sum formula
Once we know an infinite geometric series converges, we can use a specific formula to find its sum. The sum \(\textstyle \sum_{k=1}^{\infty} ar^k \) is calculated using the formula \( \frac{ar}{1-r} \). In our exercise, \( a = 5 \) and \( r = -0.7 \). Substituting these values into the formula gives:
  • \textstyle \frac{5(-0.7)}{1-(-0.7)} \
  • \textstyle \frac{-3.5}{1+0.7} \
  • \textstyle \frac{-3.5}{1.7} \
  • resulting in approximately \( -2.06 \).

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Most popular questions from this chapter

Carrie saves money in an arithmetic sequence: \(\$ 700\) for the first year, another \(\$ 850\) the second, and so on, for 20 years. How much does she save in all (disregarding interest)?

Some sequences are given by a recursive definition. The value of the first term, \(a_{1},\) is given, and then we are told how to find any subsequent term from the term preceding it. Find the first six terms of each of the following recursively defined sequences. $$ a_{1}=1, a_{n+1}=5 a_{n}-2 $$

Find the nth, or general, term for each geometric sequence. $$1,5,25,125, \dots$$

It is said that as a young child, the mathematician Karl F. Gauss \((1777-1855)\) was able to compute the sum \(1+2+3+\cdots+100\) very quickly in his head. Explain how Gauss might have done this and present a formula for the sum of the first \(n\) natural numbers. (Hint: \(1+99=100 .)\)

use the formula for \(S_{n}\) to find the indicated sum for each geometric series. $$S_{7} \text { for } \frac{1}{18}-\frac{1}{6}+\frac{1}{2}-\cdots$$
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