ImmortalFusedZamasu
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Does the following infinite series converge or diverge?

[tex] \frac{1}{2} - \frac{2}{9} + \frac{4}{27} + - \frac{8}{81} + [/tex]
A. It Diverges; it does not have a sum

B. It converges; it does not have a sum

C. It Diverges; it has a sum

D. It converges; it has a sum

Respuesta :

DeanR
The nth term is (presuming the denominator on the first is a typo)

[tex]-\dfrac{n}{(-3)^n}[/tex]

The ratio of successive terms is

[tex]\dfrac{ \frac{n+1}{(-3)^(n+1)}}{\frac{n}{(-3)^n}} = \frac{n+1}{-3n}[/tex]

The limit is -1/3, absolute value less than 1, so the series converges.  It has a sum -- that's what converges means.

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