Which of the series below converge?
[tex]$1+9+81+729+\ldots$[/tex]
[tex]$1+0.25+0.0625+0.015625+\ldots$[/tex]
[tex]$625-125+25-5+\ldots$[/tex]
[tex]$8-16+32-64+\ldots$[/tex]
[tex]$\frac{1}{2}+\frac{1}{6}+\frac{1}{18}+\frac{1}{54}+\ldots$[/tex]
Answer (2)
Identify each series as a geometric series and find the common ratio r .
Check if ∣ r ∣ < 1 for each series.
Series 2, 3, and 5 converge because their common ratios have absolute values less than 1.
The series that converge are: 1 + 0.25 + 0.0625 + 0.015625 + … , 625 − 125 + 25 − 5 + … , and 2 1 + 6 1 + 18 1 + 54 1 + … 1 + 0.25 + 0.0625 + 0.015625 + … , 625 − 125 + 25 − 5 + … , 2 1 + 6 1 + 18 1 + 54 1 + …
Explanation
Identifying the Problem We are given five series and need to determine which of them converge. To do this, we will identify each series as a geometric series and check the absolute value of the common ratio. A geometric series converges if the absolute value of the common ratio is less than 1.
Analyzing Series 1 The first series is 1 + 9 + 81 + 729 + … . The common ratio is r = 1 9 = 9 . Since 1"> ∣9∣ > 1 , this series diverges.
Analyzing Series 2 The second series is 1 + 0.25 + 0.0625 + 0.015625 + … . The common ratio is r = 1 0.25 = 0.25 = 4 1 . Since ∣ 4 1 ∣ < 1 , this series converges.
Analyzing Series 3 The third series is 625 − 125 + 25 − 5 + … . The common ratio is r = 625 − 125 = − 5 1 . Since ∣ − 5 1 ∣ = 5 1 < 1 , this series converges.
Analyzing Series 4 The fourth series is 8 − 16 + 32 − 64 + … . The common ratio is r = 8 − 16 = − 2 . Since 1"> ∣ − 2∣ > 1 , this series diverges.
Analyzing Series 5 The fifth series is 2 1 + 6 1 + 18 1 + 54 1 + … . The common ratio is r = 2 1 6 1 = 6 1 × 2 = 3 1 . Since ∣ 3 1 ∣ < 1 , this series converges.
Conclusion Therefore, the series that converge are: 1 + 0.25 + 0.0625 + 0.015625 + … , 625 − 125 + 25 − 5 + … , and 2 1 + 6 1 + 18 1 + 54 1 + …
Examples
Geometric series are used in many areas of mathematics and physics. For example, they can be used to model the decay of radioactive substances, the growth of populations, and the behavior of financial markets. Understanding the convergence and divergence of geometric series is crucial in these applications to make accurate predictions and informed decisions.
The series that converge are: 1 + 0.25 + 0.0625 + 0.015625 + \tdots , 625 − 125 + 25 − 5 + \tdots , and 2 1 + 6 1 + 18 1 + 54 1 + … . The first and fourth series diverge because their common ratios have absolute values greater than or equal to 1.
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