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Ld ; periodic block functiolairl iad Lm fW In< and f(1 = 0 for hcIl @rlodic tninelc furztion with Puml ind / 2| | for |t | g(r) Qfor 4 Show thut f1 f1is even wkn...

Question

Ld ; periodic block functiolairl iad Lm fW In< and f(1 = 0 for hcIl @rlodic tninelc furztion with Puml ind / 2| | for |t | g(r) Qfor 4 Show thut f1 f1is even wkn bolh peniodic functions with pCnOLinteznlGibhsShow" thut gvnIhe cven Denodic funclion with nennd which for " 2'<1for 0 < 6 <U +#0041Delermine Ihe Fcxlrier scnas 0l ( }+2)() and verily Ibit il converges (f#8)() foraIl( Verify MeME Fourice eccs U +8i() caleulatng tb zercxh tounetcuincient 7Uired Wily

Ld ; periodic block functiolairl iad Lm fW In< and f(1 = 0 for hcIl @rlodic tninelc furztion with Puml ind / 2| | for |t | g(r) Qfor 4 Show thut f1 f1is even wkn bolh peniodic functions with pCnOL inteznl Gibhs Show" thut gvn Ihe cven Denodic funclion with nennd which for " 2'<1 for 0 < 6 < U +# 0041 Delermine Ihe Fcxlrier scnas 0l ( }+2)() and verily Ibit il converges (f#8)() foraIl( Verify MeME Fourice eccs U +8i() caleulatng tb zercxh tounetcuincient 7Uired Wily



Answers

Let
$ f(x) = \sum_{n = 1}^{\infty} \frac {x^n}{n^2} $
Find the intervals of convergence for $ f, f', $ and $ f''. $

So to find the intervals of the intervals of convergence of a function F. Of X. The power series is and it could serve it to infinity of eggs. See the power two n plus one divided by Suen Plus one for bacteria. So let's find the interval of convergence for this our series. So I take my limits as N approaches infinity of U. N. Bloods. one divided by U. N. And this is going to be equal to my limits. As N approaches infinity I have X to the power two n plus one divided by to N blasphemy factory to two n plus three here. So three here bacteria times two n plus one facts were divided by Extra power to n plus one. Okay. And this is going to be my limits as N approaches infinity. What do I have? I have X grade. Yeah, divided by two n plus two. two n Plus three. Okay, So if you take the limit as N approaches infinity this is equal to three soon. It implies that the interval of convergence things ever of convergence is from negative infinity to positive words infinity. If we have G. So the power series G. Of S to be equal to there is any courses of infinity of my name. So on my policy was and X to the power to aim divided by to aim fact. Uh it's okay since we find a power series we take their limits as well. So the limits as an approaches infinity of U. N. bloods one divided by U. N. And this is going to be equal to our limits as N approaches infinity. What is U. N. Plus one? We have minus one. And let's one X. To the power To n. last two divided by two n. Plus two bacteria multiplied by our two in fattah divided by minus one in X. To the power sue in. Okay so this is what's how are you? A so this whole thing it's what's um then we have the U. N. Plus one. So let's simplify and this is giving us limits as N approaches infinity. We have one divided by two n. Plus two. And this will give us with zero. So same here our interval of convergence. So you know whatever interval of convergence convergence is from negative infinity to positive infinity. Then it's will let's take the derivative of F. Of S. This is going to give us the power series inequalities. Er please. So taking the face that we receive I have -1 to the power came X. To the power except power to aim divided by to a factory to so you realize that the derivative of F. Of X. Is equal to G. Of X. Then let's look at the dairy relative of G. Of eggs. This bow A series that is going to be so maybe she ain't equals zoo. So infinity. That is going to give -1 to the power in. I have soon I have eggs. The power to N -1 divided by soon in minus one fax aurora. And this is equal to My submission and equal to zero. If he needs to see Then I have my next one Step Forward and Floodators one Here. This is going to be X to the power to N last one. Okay, 10 plus one divided by two. n Plus one. Fuck too. Okay, so let's rewrite if I rewrite, you realize that this is going to give me negative my son as And called 0 to Infinity. I have -1 to the power in X. to the power to n plus one divided by two 10 plus one Fattoria. But you realize that this whole thing is towards the series of power series of words F X. Here, but there's a minus here. So this is equal to negative of words F of X. Then the nest. But so you're saying that's F F S. It's equal to same eggs, nine X. And our G of S G of S will also be equal to of course eggs because X. So if you identify the function, so you are saying that it is words. Their face was assigned function. The N G of cases also cause function. So we have been able to show that the frame of X is equal to this, then F prime of X is also equal to death. And the interval of conventions for the two power series. He's the final solution.

So before we do this question here we noticed under F and here it will be on the phone this sigh off. Thanks. Because I'm angst about stays, undersigned extent and so excited This form here and something from this function G here will be examined. That goes, I'm the axe and really the side exit interval of the convergence will be Roman infinity, infinity and something for the cosa Thanks Roman infinity, infinity and be where to show that brand of extra clearly doesn't have access And she Yanks so well if we do the amp Ram the X which again, of course, I'm the axe and the coach cheer X and something. When we have the Jeep brand X, we go do the minus side I'm the actually Could you minus f of X and that's that's a we're only figure around. This could be the function that we're looking for here


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