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#8 FIND THE INTERVAL OF CONVERGENCE of the followin POWER SERIES. OO k+z k-')k Z skki zk+2k+i K=( K=i...

Question

#8 FIND THE INTERVAL OF CONVERGENCE of the followin POWER SERIES. OO k+z k-')k Z skki zk+2k+i K=( K=i

#8 FIND THE INTERVAL OF CONVERGENCE of the followin POWER SERIES. OO k+z k-')k Z skki zk+2k+i K=( K=i



Answers

Find the interval of convergence of the series $\sum_{k=1}^{\infty} \frac{(x-2)^{k}}{k\left(3^{k}\right)}$.

Bed and take a look at this. Now, I do want to point out that everything has a K power here. Yeah. Regardless of now, there is also a number missing here, but since it was negative and it's just a constant that would have been put in there. We're just gonna just put a place where there is no one right there. Turn up. All right, So let's go ahead and use the root test. Sure. It's going to take the limit as an approaches infinity of the and through. And then of the absurd I you don't want to add over and to the end Times X -2 to the end here. So, when we take the answer out of all that and we're also taking off civil here. So this native one to the end, you don't have to worry about that. But that leaves us with X -2 -2 overhead. That's the limit as And approaches infinity. So then that's definitely gonna be equal to zero. Just always less than one. So, because of that definitely can say that this converges from like an infinity to infinity here. So that's our natural no compressions there

Get the fear where exactly this converges to here. If that and so and go ahead and take the route test. Take the lameness and approaches infinity the absolute value of so that's going to be and fruit actually right? So stand throughout all of the opposite value, wants to do that negative and to the to end Next to the two and here when we take the and throw out of that. Remember taking notes about you too. So go ahead and do that. We have negative and uh huh squared and then X to the second power. That means we take the limit as N approaches infinity of this, right? So that equals to infinity, which means it temperatures. And so if it that's the case than the only place where it converges to is that X equals here only.

And where this coverage is too. For which particular X values. Let's go ahead and use the make sure test for this. So take the limit as N approaches infinity of X to the N plus one over plus one, times 2 to the N Plus one. And multiply by the reciprocal of what we had started with so that the end times two to the end overreacts to the end here. So take the value of this year. So we we'll have here is when things simplify out here. Yeah, absolutely. Of X over two. Mr one. That means the value of X is less than two. So that means we go from negative to two. Now, I'm checking that out here. So plug in -2 into their get negative ones to the K over. Okay. And so that converges by all threatening series and then we plug into Get one over. K. Diverges by harmonic series. Then, as results, we then know that the full of convergences from negative to 22. That's right sir.

We need to find its convergence and where that's gonna be happening at. So let's go ahead and apply the ratio. Test the limit has care approaches. Mm. Brother Senate approaches infinity of 2 to the n plus one Over two and plus two factorial next to the M plus one and then multiplied by to have a pictorial over to the end actually in there. So when everything gets simplified here, this cancels to to endless too. 2 1 plus one Leaves us with the next on top and two on the top as well. and so that leaves us with two x over To end plus two Times 200 Plus one. That's the limit as N approaches infinity so that goes to zero in just less than one. Therefore our internal of convergence, it's going to be from negative infinity to infinity.


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