Determine whether the given series is convergent or divergent. Explain your answer. If the series is convergent, find its sum. ∑n=0∞3n+2n+14n

Question

Determine whether the given series is convergent or divergent. Explain your answer. If the series is convergent, find its sum.   ∑n=0∞3n+2n+14n

Mayme · Accepted Answer

Given the series: ∑n=0∞3n+2n+14n Rewriting the given series as sum of two series: ⇒∑n=0∞3n+2n+14n=∑n=0∞(3n4n+2n+14n) ⇒∑n=0∞3n+2n+14n=∑n=0∞3n4n+∑n=0∞2n+14n ⇒∑n=0∞3n+2n+14n=∑n=0∞(34)n+∑n=0∞2⋅2n4n ⇒∑n=0∞3n+2n+14n=∑n=0∞(34)n+∑n=0∞2(24)n ⇒∑n=0∞3n+2n+14n=∑n=0∞(34)n+∑n=0∞2(12)n Hence we get sum of two geometric series: r1=34 r2=12 Since both |r1|&amp;lt;1,|r2|&amp;lt;1 Both the individual geometric series converge. The given series is sum of two converging, hence the given series ∑n=0∞3n+2n+14n converges. The sum of an infinite geometric series is given as: S=1st Term1−r The sum of the series will be: ∑n=0∞(34)n=11−34 ⇒∑n=0∞(34)n=114=4 ∑n=0∞2(12)n=212=4 The sum of the given series will be: ∑n=0∞3n+2n+14n=4+4 ⇒∑n=0∞3n+2n+14n=8 Final Answer: The given series is converging. The sum of series is 8

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

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}

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