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Which of the following series are convergent?E9321318 :Vn(Vn + 1) n=1n tanne n =l 67)"...

Question

Which of the following series are convergent?E9321318 :Vn(Vn + 1) n=1n tanne n =l 67)"

Which of the following series are convergent? E9321318 : Vn(Vn + 1) n=1 n tan ne n =l 67)"



Answers

Which of the sequences $\left\{a_{n}\right\}$ converge, and
which diverge? Find the limit of each convergent sequence.
$$
a_{n}=\tan ^{-1} n
$$

We want to determine whether the sequence a N is equal to one plus seven over end all to the end power converges or diverges And if it converges, whether come bridges. So we are going to first right down our limits of the limits. As an approaches, infinity of a N is equal to be limits, as in approaches infinity of one plus seven over end to them. Now, this is going to be a straightforward application of here in five. So this is going to equal e took seven. Bye here. Room five number five Sense X is equal to three, so this function converges and a converted C E to be seven.

We get to determine if this Siri's convergence or the Gerges in this case, we're going to use a ratio test to prove that the Siri's they vergis. First, let's see that the term of the serious ace and N is equal to that. They want to. The nth power times three in factorial, divided by in factorial times, endless one factorial times endless, too Centauri and Use Ratio Test. We got to find a limit off the expression absolute value of a PSA endless one divided by a sub in, and this is equal to the absolute value off. A simple swan is negative. Want to the endless one times three times Simples Juan pictorial divided by and plus one factorial times and plus two factorial times and plus three factorial. And that divided by a Superman, which is equivalent to multiply by the inverse of face of end, which is in factorial times and plus one factorial times M plus two factorial divided by native want to the nth power Times three and factorial and this sequel to three n plus three, which is this expression here making a distribution of the multiplication through the sun. So we get three births, three employees, three factorial and he's going to hear negative. Want to the M plus one. Absolute values. One. So we have these one times in factorial, which is this one here Next M plus one factorial and at times, empress to federal. That's the numerator and denominator is Empress one factorial times and plus two factorial times AM plus three factorial Absolute value of Nancy Want to the end? It's one so times in three grand for group. Now we can roll the factorial three end for 32 times to get three in plus three times three n Bless too Times three in plus one times three n factorial We do that to have the same three in Victoria in the Numerator and Denominator Time. Same factorial times in factorial times and plus Juan Factorial time simplest to factorial divided bye M plus one factorial times simples to factorial Times employees three factorial times three in furter And now we can't simplify three In factorial we can simplify also m plus one factorial here in M plus two factorial. So we and we end up with sequel to this fraction, which is three m plus three trembles through the implicit one. Three Chambliss. Three three endless, too three and plus one times in factorial. That's in the very torrent. The number nature we have Implant three from Factorial M plus three. Factorial. What and no, we may way make the same thing in the denominator. Unrolled the factorial just to have in factorial. So we have. And in the first factor here three. In 43 we take out three common factors. Always. Three M plus one time. Three n plus two times three In blood Swan times in factorial Divided by In Here in the Terminator We have employed three times m plus two time simples. One time, same factorial. So now a factorial. Can we simplify and endless one here in here? So three get finally three times a three n plus two time three and plus one, divided by M. Bliss three times and plus two. And she's equal to three times three and two and three, and he's nine and square plus three in time. Sworn in, three in and two times 3 86 and so is nine in Bless Too divided by. And here we have n square. Uh, bless two and a three and five in plus three times to six. So this is equal to 70 37 square, plus 27 plus six the wind and by in square plus five and blessings, which is equal to 27 plus 27. What about Anne? Plus six, divided by Chan and square is obtained by dividing the numerator by and square is the same for the denominator. We God one blast five divided by and plus six divided by the square. In With that, we know that the limit when and goes to infinity off the absolute value off a sin and plus one divided by a N is equal to the limit of this expression here that is going to be 27. Developer wine, that is 27. She's creator, that one. So the Siri's they've urges

We want to determine whether this sequence day and is able to negative one to the end plus one times in post war over and converges for diverges. And we want to give reasons as to why so First, I'm just going to rewrite impulse one over in as so they want to the n plus one and dividing the end And teacher those we get one plus one over. Now notice that fifth in is even. So I'm just like this. So if n is even, this sequence is going to be negative one to an even power. So just say to m plus one, one plus one over two him now negative 1 to 8. Even power will just be one. So we end up with 21 plus one over to him and then we know that one goes to one and two over. M goes to zero. So we just end up with this going to two as M goes to infinity. But when n is odd, well, we're gonna have negative warned to the to M plus one plus one all over one plus two plus one. Well, negative one toe. A negative power will be negative one and the negative one +10 So we'd end up with zero times one plus one over two plus one, which is zero and also so. As M approaches, Infinity is also goes to zero Now because we have two different sub sequences here that approach different things, we can say that this will diverge. So die Burgess since not all some sequences approach the same limit.

This is our question over 92 Here our sequence is 11 square two turning waas. And so here we have multiplication or one Do you like 10? So we can write 10 Denominator lying to me Today's limit and screwed. Now execute and instant finance your finite Don't go right training yours in. Fine it is Bye bye to and unloading. Fine it, ese. So here. Bye bye to restaurant. Finite is CEO because you're one divided by infinities also Zero. So your limit is two received policies Ridges CEO. Thank you.


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