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Problem 38 Hard Difficulty

If $ f(x) = \sum_{n = 0}^{\infty} c_n x^n, $ where $ c_{n + 4} = c_n $ for all $ n \ge 0, $ find the interval of convergence of the power series and a formula for $ f(x). $

Answer

$f(x)=\sum_{n=0}^{\infty} c_{n} x^{n}=\frac{\left(c_{0}+c_{1} x+c_{2} x^{2}+c_{3} x^{3}\right)}{1-x^{4}}$
The convergence radius is $R=1,$ with interval of convergence: $(-1,1)$ .

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Video Transcript

for this problem. It really helps toe. Just write out the first few terms here. So we start out with C zero times X to the zero X zero is one. And then we have plus the one times X plus C, too Times X squared plus C three, Tim's X cubed. And then the next term we have is C four times extra before. But this equation here tells us that c for the same thing as zero. So now we just have C zero exit the fourth, and then similarly, over here we would have C five, but see, five is something as C one according to this equation, Now we have plus C one X to the fifth and similarly over. Here s sorry about the bad handwriting and the pricey to pattern now, So we congrats together. All of these C zero terms group together all the C zero terms. We're starting with something that has exploded zero. But then eventually we see that the exponents and acts are going to be increasing by four. You put a four here, put it in here and then similarly, when we look at all the terms that have C one is a coefficient We'LL have the same type of thing here except we'LL have an extra power of X Now we have C one times X and the same same type of thing skin so you probably see the pattern there. So we can we see that all of these all the terms here gonna have this common factor of this sum here We can factor out this some that we see happening here When once we factored out who solved the c zero to take care of? We have the C one acts and then off screen we would have C two X squared plus C three x cube. Okay, So if we're trying to figure out the interval of convergence, the only thing that we have to worry about is this some here. And if you do the ratio test to this sum, you'LL see that what you end up having to get is that the absolute value of X of the fourth is less than one. And for that to happen, we would know that X has to be between minus one and one. So that means that all we have to figure out now is whether or not we include minus one and whether or not we include one in R sum. Okay, but if we plug in minus one or one into here, then we're just gonna be summing up one an infinite amount of times. That's clearly not gonna work. So we'LL toss out minus one and one for a couple of convergence and leave this as an open some here. So that's our interval of convergence. And now get the closed formula. Recall that this has a closed form one over one minus X, and we're working with something similar here. Instead of acts, we have extra the fourth. So now we just use this formula here except replace our X with X to the forth, and we get one over one minus next to the fourth. And we can't forget all the stuff that we're multiplying here. So we had this c zero plus XY one plus X squared C two plus X Cube C three and we're multiplying by this some, which is something. It's just divided by one minus X to the fourth. That's the clothes form. And just as a reminder, The interval of convergence, we said, was minus one toe one throwing out minus one and throwing out one