Question

Determine whether the sequence converges or diverges. If it converges, give the limit.

48, 8, four divided by three, two divided by nine, ...

Converges; two hundred and eighty eight divided by five

Converges; 0

Diverges

Converges; -12432


Help me please show work and explanation

Answer 1

Answer on Question 42507, Math, Calculus

It is obvious, that the given sequence might be rewritten as $2^{4} \cdot 3^{1}, 2^{3} \cdot 3^{0}, 2^{2} \cdot 3^{-1}, 2^{1} \cdot 3^{-2} \ldots$ . Thus, general formula for $n$th term is $a_{n} = 2^{4 - n} 3^{-(n - 1)}$ , or $a_{n} = 48 \cdot \frac{1}{6^{n}}$ . This sequence converges, because it is a geometric progression with $q = \frac{1}{6}$ , multiplied by 48. Thus, the sum is

$$S = 48 \cdot \frac{1}{1 - \frac{1}{6}} = \frac{288}{5}.$$

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