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does the following infinite series converge or diverge? 1/3+2/9+4/27+8/81+...
a. it diverges, it has a sum
b. it's converges, it has a sum
c. it diverges, doesn't have a sum
d. it converges, doesn't have a sum

Respuesta :

LammettHash
Note that A and D are ludicrous choices, so you can throw them away outright. (Any divergent series cannot have a sum, and any convergent series must have a sum.)

The sum is certainly convergent because it can be written as a geometric sum with common ratio between terms that is less than 1 in absolute value.

[tex]S=\dfrac13+\dfrac29+\dfrac4{27}+\dfrac8{81}+\cdots[/tex]
[tex]S=\dfrac13\left(1+\dfrac23+\dfrac{2^2}{3^2}+\dfrac{2^3}{3^3}+\cdots\right)[/tex]

We can then find the exact value of the sum:

[tex]\dfrac23S=\dfrac13\left(\dfrac23+\dfrac{2^2}{3^2}+\dfrac{2^3}{3^3}+\dfrac{2^4}{3^4}+\cdots\right)[/tex]

[tex]\impliesS-\dfrac23S=\dfrac13[/tex]
[tex]\implies\dfrac13S=\dfrac13[/tex]
[tex]\implies S=1[/tex]

So the answer is B.

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