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Consider the series 1/4 + 1/6 + 1/9 + 2/27 + 4/81 +....

Does the series converge or diverge?

Select answers from the drop-down menus to correctly complete the statements.

The series __________ (converges or diverges). You can conclude this because the series is _______________ (arithmetic. geometric and the absolute value of the common ratio is greater than 1. geometric and the absolute value of the common ratio is less than 1. or neither arithmetic nor geometric.)

 Jun 9, 2021
 #1
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Key idea:

1) If the given sequence does have common difference, or common ratio, then it is neither an arithmatic nor geometric.

2) How to find out :

We say "d" is common difference of arithmatic :

If d value is   (a2 - a1) ≠ (a3 - a2 )  then it is not arithmatic sequences.

 

We say "r" is common ratio of geomatric sequences:

If r value is  (a1 / a 2) ≠ (a2 / a3 ) , then it is not geomatric sequences. 

...  Here, we get common ratio of geomatric sequence =  (1/4) /(1/6) = (1/6) /(1/9) = (3/2)  , Then it is geometric sequences.

3) Is  (3/2) greater  than 1?

4) If geometric common ratio  is greater than 1 , it means the sum of the series  keep increasing , then we say it is diverges.

 Jun 10, 2021
edited by Bginner  Jun 10, 2021

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