SOLUTION: Consider the infinite geometric series (linked below). In this image, the lower limit of the summation is "n=1". As I am not sure how to type this in formatting on this website

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Question 1117305: Consider the infinite geometric series (linked below). In this image, the lower limit of the summation is "n=1".
As I am not sure how to type this in formatting on this website or how to type this in text this is a link to an image of the formula mentioned above.
https://postimg.cc/image/hchcwn7jr/
A) Write the first four terms of the series.
B) Does the series diverge or converge?
C) If the series has a sum, find the sum.
Found 2 solutions by rothauserc, solver91311:
Answer by rothauserc(4718)   (Show Source): You can put this solution on YOUR website!
A) -4, -4/3, -4/9, -4/27
:
Note the first term is -4 and the common ratio(r) is 1/3
:
B) converges since |r| < 1
:
C) -4 / (1 - 1/3) = -4 * 3/2 = -6
:
Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!


sum[i=1,infty] [-4*(1/3)^(n-1)]
is an understandable plain text representation of





As to convergence/divergence, there are a couple of ways to look at it.

You can write



But is a geometric series with . Since any geometric series with converges, this series must converge.

Alternatively, you can use the Series Ratio Test: A series where:



where converges. Verification of the series ratio test is left as an exercise for the student.

Either way, the series converges.

The sum of a geometric series with a common ratio is given by:



So



You can verify the arithmetic for yourself.

John

My calculator said it, I believe it, that settles it
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