SOLUTION: In the convergent series, 5 -10/3 +20/9 -40/27... what does the series converge to? The answer is one of the following. Which one? A) the series converges to 5 B) the seri

Algebra ->  Sequences-and-series -> SOLUTION: In the convergent series, 5 -10/3 +20/9 -40/27... what does the series converge to? The answer is one of the following. Which one? A) the series converges to 5 B) the seri      Log On


   

Question 1198628: In the convergent series, 5 -10/3 +20/9 -40/27... what does the series converge to?
The answer is one of the following. Which one?
A) the series converges to 5
B) the series converges to 15
C) the series converges to 3
D) the series converges to 9
Found 2 solutions by MathLover1, math_tutor2020:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
The nth term of this sequence is

a%5Bn%5D=-%2815%2F2%29%28-2%2F3%29%5En

sum%28-%2815%2F2%29%28-2%2F3%29%5En%2C+n=1%2C+infinity%29+=+3

C) the series converges to 3

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Answer: 3 (choice C)

Explanation:

Refer to this previous similar question.
https://www.algebra.com/algebra/homework/Sequences-and-series/Sequences-and-series.faq.question.1198622.html
In it I discuss that this sequence is geometric with
a = 5 = first term
r = -2/3 = common ratio

I then further explained that
S = a/(1-r)
S = 5/(1-(-2/3))
S = 3
represents the summation of the infinitely many terms.

The formula
S = a/(1-r)
only works when -1 < r < 1 is true.
Otherwise, the geometric series would diverge.

Luckily -1 < -2/3 < 1 is true, so -1 < r < 1 is true for r = -2/3.