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## Homework Statement

"Determine whether the following series converge or diverge. If the series is geometric or telescoping, find its sum.":

## \left ( \sum_{k=1}^\infty2^{3k} *3^{1-2k} \right)##

## Homework Equations

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The different tests for convergence?

## The Attempt at a Solution

Ok, I've looked at all the tests for convergence I know, and as far as I've been able to tell, none of them can work with this? Here's a quick outline about each of the tests and why I don't think I can use it.

*Divergence Test:*I have to say I can't even do the limit of this. It just gives me ##2^{infinity}*3^{1-infinity}##, no?

*Geometric series*: I don't have only one exponent, so I can't do this. Neither of the exponents looks like ##(n-1)## so I don't know.

*Telescoping series:*There's nothing that cancels out. I also can't take the limit.

*p-Series*: It's not in the form ##{\frac {1} {p}}##.

*Alternating series:*It doesn't have a ##-1^n## term.

*Integral test*: ##\int 2^{3k} *3^{1-2k} \, dx## . I can't integrate this because of the ##k## being a power.

*Root test*: While both are raised to a power of ##k##, I don't think I can apply the root test because of the ##1-2k##.

*Ratio test:*This is usually my go to, and I thought it would work. However, I get ##\frac {1} {9}## and then the ##3^{1-2k}## just stays there.

*Comparison test*: Even if I wanted to use the limit or direct comparison, I wouldn't even know what to compare it with.

The solutions tell me it's supposed to converge to 24 and to use the geometric test. However, I really don't see how the geometric test is supposed to apply.

Thanks for your time, and I hope I did everything right in posting this!