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The interval of convergence of the power series Xko %i is 00 7 [-1,1] (-1,1) 0 0 0 0 (1,0) (To,x)...

Question

The interval of convergence of the power series Xko %i is 00 7 [-1,1] (-1,1) 0 0 0 0 (1,0) (To,x)

The interval of convergence of the power series Xko %i is 00 7 [-1,1] (-1,1) 0 0 0 0 (1,0) (To,x)



Answers

Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $$\sum_{n=0}^{\infty}\left(\frac{x}{k}\right)^{n}, \quad k>0$$

Middle of conversions for this power series do that use the ratio test first taking the limit as n approaches infinity of two x to the this one By two x 2 X. to the power here. This gives us two x times the value of two X is less than one year. So then got that supply of X is less than white house. So then we have from negative one half To 1/2 as a potential. We need to test the endpoints so let's go ahead and try that up. We could see at X equals negative one half. We get -1 to the head power which the bridges then at X ankles one half, get onto the and power it's also and I purchase so therefore her interval convergence from negative one half, 2.5.

The interval of convergence. Let's go ahead and apply the ratio. Test to find out, ignore the negative ones at the end because we take the absolute value anyway, It's going to be exited to end plus one Got to end plus two rather Owner and plus one factorial times and factorial over next to the to end. So that simplifies to X squared over and plus one and the limit as an approaches infinity, She goes to zero, Which is always less than one. So therefore it comm purchase from negative infinity to infinity and that's our interval of no fragrance.

In the interval of corporations to do that. We have to use the ratio. Test the limit as N approaches infinity and ignore the negative 12 then plus one because we're taking the absolute value anyway. So it's x minus one to the helpless too overhand. Plus two it's reciprocal inches, N plus one over X -1 to the and Plus one. Right. When everything is said and done we take the limit to infinity. The ends here go to one. The extent most one to the M plus one cancels out. And so we're left with the experienced one. So we had this being less than one. So we've got the interval being from uh negative one X minus 1 to 1. Anyone to both sides. We have between cereal and to and now we need to test the endpoints. So testing X equals zero. We have -1 to the end plus one Times -1 to the n. plus one here. All over and plus one. And so that cancels out the negative portion on top, Which leaves us with one over. n plus one. Which type purchase. Uh huh. Bye. Because it's we can say that diverge per P. Series or direct comparison to P series If we take a look at the next one so at x equals two -1 to the end plus one times one Over and Plus one. And this word convert by alternating series. So therefore our interval of convergence, It's going to be from 0 to 2 with the bracket at two because it's included.

And the internal convergence. Do that. 1st. Take the limit as approaches infinity. We're doing the ratio. Test this. So we have data want and plus two plus two times extend that plus one Divided by a -1 to the plus one Krampus one. To the extent that here and so supplying this here, we can ignore the negative ones because that's absolute value. The endless to an endless one. Go to one when we take into infinity. And the extra bands here pieces with just X. So our potentials from negative one 21 But we need to test the endpoints here. So let's take a look at that. You look at x equals negative one. We get -1 to the end plus one Times and Plus one times negative 1 to the head. So this will diverge. Sure, excellent. To that just goes to infinity. Just to keep going and going and going. If we take a look at X equals one -1 to the end plus one Some Plus one times 1. It also diverges because that's also getting bigger and bigger, Right? So that means our interval of convergence. That's your negative 1- one.


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