Given:
Given the series
[tex]\frac{1}{5}+\frac{1}{10}+\frac{1}{20}+\frac{1}{40}+...[/tex]Required: Identify the following series as geometric or arithmetic. Also identify the series as infinite or finite.
Explanation:
The given series is a geometric series as the terms of the series are 1/2 times its previous term.
[tex]\begin{gathered} a_2=\frac{1}{10}=\frac{1}{2}\cdot\frac{1}{5}=\frac{1}{2}a_1 \\ a_3=\frac{1}{20}=\frac{1}{2}\cdot\frac{1}{10}=\frac{1}{2}a_2 \\ a_4=\frac{1}{40}=\frac{1}{2}\cdot\frac{1}{20}=\frac{1}{2}a_3 \end{gathered}[/tex]So, the given series is a geometric series with first term, a = 1/5 and common difference, d = 1/2.
The three dots at the end of the series means the series is continuing. So, it is an infinite series.
Final Answer: The given series is an infinite geometric series.
