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25.) The p-series 2 diverges if psi diverges if p > [(b) diverges if p <1diverges il pzi(e) None ol these...

Question

25.) The p-series 2 diverges if psi diverges if p > [(b) diverges if p <1diverges il pzi(e) None ol these

25.) The p-series 2 diverges if psi diverges if p > [ (b) diverges if p <1 diverges il pzi (e) None ol these



Answers

If $0< a_{n}$ $\leq b_{n}$ and $\sum_{n=1}^{\infty} a_{n}$ diverges,then $\sum_{n=1}^{\infty} b_{n}$ diverges.

In this exercise, we come to the side if this statement is true. So this tainment looks very similar to the comparison Direct comparison test. But it's a slightly different. So he's saying that if a n it's a smaller regaled and being and be in d Burgess diver, choose then also a in my burgers. So in the case off the direct comparison tests, where we have is that bnb? Yeah, converges then a convergence. This is slightly different. And the truth is that this statement isn't five in fact, false. Okay, let's see an example. For example, if you consider a and to be one of Ren Square one over and square is the convergence sequenced in and be over in to be won over an isa divergent series. Now siro is always bigger required on one over end square that is so is smaller equal than one program. So this is a counter example because we know that the serious being diver just and siris a n converges

So as a Ben is equal to e to the power of to and over and plus to power. So we're gonna be evaluate this limit okay of the sequence. So what you guys will do is you guys wanna look at this power over here, so limit if you guys can evaluate this power, um, then you guys should be able to find the whole limit. Okay, So limit and goes infinity. Uh, Tuesday event, a two event over and plus two. And this is going to be one. Um, you guys can divide this guy by end by describe the end, and this will give us what limit and goes infinity too. One plus two over and and this whole thing goes to zero. So what do you have left? Limit and goes infinity. That's to over one. And this is going to be too. So the sequence will be converged going to e to the part that we just evaluate the power. Okay, So the limit and goes infinity is event is equal to limit and goes infinity, uh, to the power of to So this is gonna be e square. And please be careful

Ace events. So you got to eat department plus E to the minus and over. Eat part of two s a van minus one. And you guys want to do this e to the power of to end. And you guys wanted to buy this guy, by the way, To the part twice. Okay. Okay. And also for here. Okay, Now, what happens is after this, let's put it in the mid for the limit and the approaches Infinity, uh, eat to prevent baba e to the power to end. That will be just one over. Eat the end. Right? And plus eaten minus and baba eat the to end. There'll be just one over. Eat the power of three end, isn't it? Right? Okay. And yeah, this over. Eat the twice event over that one. Minus one number. Eat the part twice. Event. Now, this Each fraction part over here approaches to zero. Okay, everything becomes zero. So here, zero plus 0/1, and this is gonna be just zero. So this sequence converges to zero

For her project appears to see whether this series is absolutely convergent, conditionally convergent or temptations. And so let's go ahead and use the root test here. Take the limit as N approaches infinity here, the entire route of and we're going to take the opposite value of negative to end The five and power overhand plus one The five and power as well. All right. So when we're taking a look at this here, so after we take the threat of that absolute value of all is here, then we're end up with to end To the 5th power porter. Yeah, actually just okay. And then all that over And plus one fifth power here, take the limit as N approaches infinity here. And so what we'll see here is that we've got 32 and to the 5th over into the fifth and then so on and so forth. When it's expanded here and take the limit as N approaches infinity here, That equals to 32 since the powers here are the same. And so You see that that's greater than one. Therefore, by the ratio test we know that this series diverges


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