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WZ
Numerade Educator

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Problem 22 Medium Difficulty

Find a power series representation for $ f, $ and graph $ f $ and several partial sums $ s_n(x) $ on the same screen. What happens as $ n $ increases?
$ f(x) = \ln (1 + x^4) $

Answer

$$\ln \left(1+x^{4}\right)=\sum_{n=0}^{\infty}(-1)^{n} \cdot \frac{x^{4 n+4}}{n+1}$$

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Video Transcript

The problem is find a power serious representation of your half and the graph on several partial psalms on the same screen. What happens as any increases. So first one over one plus X is equal to one over one madness. Negative attacks, which is you got you some. I'm from zero to infinity. Negative X to the power off in your after a while You fax is fast. What? So out in Juan plus tax You can't you into girl off one over one plus X which is equal to some problems, you know, three vanity next you'LL want to the part ofthe end halves X to the power plus Juan over and plus plonk Last constant number. See, we put X zero in this equation so we can find a C is equal to help and one plus zero which is equal to zero. So we have Ellen one plus X. It's like watching some from zero to infinity Next one to the power ofthe end house X to the power ofthe plus one over and plus one. Then we replace axe exc the power of floor so half hour how in one class X to the power off? Or is this going to sound from zero to infinity? Make you want to the power off in terms Axe to the powerful and last four over and plus one. Yeah, absolute value ofthe ACS is the last one. This is a box. Now let's look at this graph over after rocks. No, we can't not the graph off as one as two, Andi asked. Really the paras arms, as one has to ask three. Notice that as an increases as an ax becomes oblige approximation to the Apple Jacks Well, acts between make two one on one.