Determine whether the following series converges or diverges. ∑n=1 to ∞ 6/(9+9^n)

Question

Determine whether the following series converges or diverges. ∑n=1 to ∞ 6/(9+9^n)

Answer 1

By Applying Series Ratio Test -> CONVERGES

Z = lim k -> inf [Uk+1] / [Uk]

Z<1 converges

Z>1 diverges

Z=1 inconclusive

Uk = 6/(9+9^k)

Uk+1 = 6/(9+9^k+1)

[Uk+1] / [Uk] = (9^n +9)/9(9^n +1)

Z = lim k -> inf [(9^n +9)/9(9^n +1)]

Z = 1/9

Z<1 true thus converges

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NOTE

Series converges at the condition such that

1 / 9^k + 9 = 0 ; k element of Real

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Answer 2

Because 0 < 6/(9 + 9ⁿ) < 6/9ⁿ and series ∑₁∞6/9ⁿ converges as geomotric series with first term a₁ = 6/9 = 2/3 and common ratio r = 1/9 and - 1 < 1/9 < 1. So original series by comparisson test also converges.