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JH
Numerade Educator

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Problem 45 Medium Difficulty

(a) Show that $ \sum_{n=0}^{\infty} x^n/n! $ converges for all $ x. $

(b) Deduce that $ lim_{n \to \infty} x^n/n! = 0 $ for all $ x. $

Answer

a. $\sum_{n=0}^{\infty} \frac{x^{n}}{n !}$
b. the series of part (a) always converges, we must have $\lim _{n \rightarrow \infty} \frac{x^{n}}{n !}=0$ by Theorem 11.2 .6

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Video Transcript

Let's show that this Siri's converges for all ex. Some party, eh? With use the ratio test for this is our r A n So the ratio test for requires us to look at a N plus one over a end absolute value of that and then take the limit is and goes to infinity. So here will have so that a m plus one of the top and then they end in the bottom. Now we should go ahead and flip that red fraction over and multiply dividing by a fraction here. So we'LL have X and plus one over X n and then in factorial over and plus one that's for real. So here's the key observation here. You just look at this fraction. So using the definition of the factorial function, she write that out, You could see that we could cancel the first and batters and you just slept over with one over and plus one. So here will have just one ex leftover eggs on top and plus one in the bottom. But here, for any real Number X that we have excessive fixed number, it's not a limit here. The end is in the limit, and this will always go to zero due to that denominator. So you could think of this is being X over infinity. But X is not infinity or minus Infinity X is a real number, so we have zero, which is less than one. So we conclude that it converges for all ex, and that was the majority of the work here because part B will follow from the test for divergence. So this is a corollary of the test for diversions. So sorry here, mixing on my letters. The test for diversion says if the Siri's converges, then the limit of the and has to go to zero. So that's just the theorem in the book. But in our case, the limit of a end based on our definition of an this is just a limit as n goes to infinity X to the end over and factorial. So you go back and erase that. That's and factorial there kind of running out of room here. Scylla meeting There's my in factorial and then by the serum above that were citing, This has to be zero and that our deduction in part B and that completes our answer