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E1or(8 ~pts) For the power series , find the [dius and interwal of convergence (t'( 2" "2"...

Question

E1or(8 ~pts) For the power series , find the [dius and interwal of convergence (t'( 2" "2"

e1or(8 ~pts) For the power series , find the [dius and interwal of convergence (t'( 2" "2"



Answers

Find a power series representation for the function and detemine the interval of convergence. $$f(x)=\frac{x}{2 x^{2}+1}$$

Right we want to find the convergence set that is all values of X for which the given power series converges where the series is the sum from N equals one to infinity of ex soviet over M squared. This question is challenge your understanding of power series in particular how to find a conversion set for experience the end term of this series as X city and over em square. We can use the term in conjunction with the absolute ratio test to identify where this series converges. So first for the absolute ratio test we find P which is the limit as N approaches infinity of absolute value A N plus one over am plugging in N plus one end for our given term gives absolute value X. The N plus one over N plus one squared divided by X 30 and over em square. This is absolute value X squared over N plus one square taking limited and approaches infinity. This is simply absolute value X. R series converges when absolute value P is less than one. So we are converging starting from negative 1 to 1. Since both endpoints converge, We have negative 1-1 inclusive.

In this question. We have a function once upon two minus X. We need to find a dramatic power theories for this function and we need to find its interval of convergence. So let's see how to solve this question. This function can also be written as one upon one minus x minus one. We know that if D sequence oh our shells um has he limit as and tends to infinity, then we can conclude that the cities converges otherwise diverges. And we know that if the common ratio of a geometric series is less than one, then the formula to calculate the sum of geometric cities can brightness as difficult to first term a divided by 1 -3an ratio are no compare this formula with the given function. So we get yeah first term is equal to One and the common ratio R is equal to X -1. We know that mm cities converges if Common ratio R is less than one. Therefore we can conclude that X -1 is less than one hands. X will lie between zero and two. So this is the interval of convergence. And now let's find the geometric cities. We know that the general expression for the geometric series is given by as his equals to pastor A plus A R plus E, ari Squire, plus A. R. Q plus so on. Now substitute all the values. So we get thumb as it recalls to one plus one multiplied by X -1 plus one, multiplied by x minus one to the power to plus one multiplied by x minus one to the power three plus so on. Up to infinity. Hence finally, we get the cities As it recalls to one plus, explain this one Plus X -1 to the power to plus so on. Plus The connection will be called to X -1 to the power and plus so on. Up to infinity. So this is the final answer for this problem. I hope you understand the solution. Thank you.

Okay, so we have this Some here, right? One plus explains to x minus two. Squared plus X minus two cubed. Plus that plus X minus two to the K. We can go ahead and we can rewrite this as some where we have K going from zero to infinity of Well, just X minus two to the cake. Okay, So we have here, then is a geometric Siri's where we have a being equal to one, and we have our being equal to X minus two. This serious conversions when the absolute value of art is less than one. So we have the absolute value of X demise. Two is less than one. Where should we have? Negative one is less than X minus two, which is less than one, which means that we'll negative one plus two is less than X minus two plus two, which is less than one plus two. Therefore, we haven't won is less than acts, which is less things. Okay, um so therefore, the interval of convergence is the interval. While the opening about 1 to 3 letter for that implies that one is less than acts, which is less than three. Okay, So there's our interval of convergence now. Are some here? Well, is equal to a over one minus R, which is equal to well, 1/1 minus X minus two. Okay, which we have. Well, that's equal to one Over, um, one minus X. Um, plus two, which is gonna be 1/3. Minus acts. Right? So there's our familiar What digger?

So fun houses the reputation for the function FX and determine the in the role of convergence. So if x equals, it's worth us experts eight times expire and Ace is a positive number. So this is our way of power. Siri's. So this is a polynomial, and we fund that it has. So the coefficient of X to the power zero is a square And that sit that of the ex term, this one and that if the Expressionists eh? And so awful Intermix zero zero zero zero zero. So that is our area power Siri's. So this is thie code revisions. That's her so efficient that way yet So it's a royal power. Siri's on determine the winner will pin murders on DSO FX. We'LL always converge to itself. So we're going to see the honorable two murders. Is thie whole real lion. So is out of the room number


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