Given the geometric series sum_{n=2}^inftyfrac{7(3)^n}{(-5)^n} a) Find r, the common ratio. b) Determine if the series converges or diverges. c) If it converges, find the limit.

Wierzycaz 2021-01-31 Answered
Given the geometric series n=27(3)n(5)n
a) Find r, the common ratio.
b) Determine if the series converges or diverges.
c) If it converges, find the limit.
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Expert Answer

Velsenw
Answered 2021-02-01 Author has 91 answers

Given geometric series is n=27(3)n(5)n
(a) To find the common ratio:
Consider the series, n=27(3)n(5)n
First term is 7×(35)2=6325
Second term is 7×(35)3=189125
Common ratio r=(7×(35)3)(7×(35)2)=35
(b) To determine the series is convergent or divergent:
Here common ratio is 35
|r|=|35|=35<1, thus the series is convergent.
(c) To find the limit:
The limit is same as that of the sum of the series.
Here n=27(3)n(5)n is a geometric series with common ratio 35, also which is an infinite series.
Sum of an infinite series is a1r, where a is the first term and r is the common ratio.
In the given series a=7×(32)2,r=35
Therefore the sum is (7×(35)2)1(35)=(7×(35)2)1+35=6340
Thus, the limit is 6340

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Jeffrey Jordon
Answered 2021-12-27 Author has 2047 answers

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