Problem 39 Hard Difficulty

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

Answer

Watch More Solved Questions in Chapter 11

Video Transcript

problem is latte after Max is a or two some and from one to infinity, extra end over and squire finding the intervals of CA merchants or half of prom on DH f from from So first doing to find this avoiders of convergence. So for this function, use thanks swung over us. One squire over acts to end over back on square is absolutely lim and cost to infinity. This is unconscionable. Absolutely off extra and plus one over. I'm a plus one squire times and squire overact strip and absolutely Oh, lemon, I meant cost. Three vanity you work in concert has to end here. This is Max. The limit is a call to outside while you off Max by ritual test If this is Russell one the function of a box as comm Ardent. Absolutely, absolutely So the readers of convergence is a cultural one. Then we check and points next to one on one. What function have Lux What next is equal to one like the next to one bucks, as they call to some from one to infinity. Next to want to end over and square at one x is equal one. Lux. It's a call to some from blind to infinity wand over and squat face Here, Power off end is, too, which is greater than one. So after box, this comer absolutely so also some murders. So for works Interval of Convergence, it is rocket next to one and one for us. Prom. We're half providers of convergence is also one. And then with tax and the points What ex after wax from Mexico to song from one to infinity. Here this is axe to once one over and it's from like to one If they caught you song from one to infinity, nine to one to support off my one over in. By loving this test, the Siri's murders, it's off from that's point one, this physical to some one over M birdies since the power off and as he could, one, which is less than one hemorrhages. So for ex prom Interval of Convergence is Brockett next to one one. Princesses, well sac derivative, the inter radius of CA murders. It's also one way also need to talk. It's in the points, so first ask from from as a control some I'm strong to infinity Axe too modest to over and Toms and minus one. So because after prom prom, that's point toe. Make two one. Is this going to some? Next to Juan comes a month, one back to one, to the parlor, off on my list, too times a month, one over then. Since the limit off the general term is not equal to zero, the limit of this line is not equal to zero. So this Siri's now where it is the thin reason we can see. Ask Trump Trump that's appointed one, also a murderess. So for the function of axe from prom, interval of components is Princess's neck to want one.