Question

Does this Series Converge or Diverge? What is the common ratio?

Answer 1

Answer:

Converge

Step-by-step explanation:

Given geometric series

$1,000+200+40+8+\dfrac{8}{5}+...$

The first term of this geometric series is $a_1=1,000.$

Next terms can be obtained by dividing by 5 or multiplying by $\frac{1}{5}:$

$a_2=1,000\cdot \dfrac{1}{5}=200\\ \\a_3=200\cdot \dfrac{1}{5}=40\\ \\a_4=40\cdot \dfrac{1}{5}=8\\ \\a_5=8\cdot \dfrac{1}{5}=\dfrac{8}{5}\\ \\...$

The sum of this infinite geometric series is

$S=\dfrac{a_1}{1-q}=\dfrac{1,000}{1-\frac{1}{5}}=\dfrac{1,000}{\frac{4}{5}}=1,250$

Hence, this series is convergent