Determine whether the given series is convergent or divergent. Explain your answer. If the series is convergent, find its sum. sum_{n=0}^inftyfrac{3^n+2^{n+1}}{4^n}

generals336 2020-10-25 Answered
Determine whether the given series is convergent or divergent. Explain your answer. If the series is convergent, find its sum.
n=03n+2n+14n
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Expert Answer

Mayme
Answered 2020-10-26 Author has 103 answers

Given the series:
n=03n+2n+14n
Rewriting the given series as sum of two series:
n=03n+2n+14n=n=0(3n4n+2n+14n)
n=03n+2n+14n=n=03n4n+n=02n+14n
n=03n+2n+14n=n=0(34)n+n=022n4n
n=03n+2n+14n=n=0(34)n+n=02(24)n
n=03n+2n+14n=n=0(34)n+n=02(12)n
Hence we get sum of two geometric series:
r1=34
r2=12
Since both |r1|<1,|r2|<1
Both the individual geometric series converge.
The given series is sum of two converging, hence the given series n=03n+2n+14n converges.
The sum of an infinite geometric series is given as:
S=1st Term1r
The sum of the series will be:
n=0(34)n=1134
n=0(34)n=114=4
n=02(12)n=212=4
The sum of the given series will be:
n=03n+2n+14n=4+4
n=03n+2n+14n=8
Final Answer:
The given series is converging.
The sum of series is 8

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Jeffrey Jordon
Answered 2021-12-17 Author has 2262 answers

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