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WZ
Numerade Educator

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Problem 85 Hard Difficulty

Use the following steps to prove (17).

(a) Let $ g(x) = \sum_{n = 0}^{\infty} (^k_n) x^n. $ Differentiate this series to show that

$ g'(x) = \frac {kg(x)}{1 + x} -1 < x < 1 $

(b) Let $ h(x) = (1 + x)^{-k} g(x) $ and show that $ h'(x) = 0. $

(c) Deduce that $ g(x) = (1 + x)^k. $

Answer

a. $\frac{k g(x)}{1+x}$
b. 0
c. $x \in(-1,1)$

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Video Transcript

the problem is use the following steps to prove 17 17 is one plus X to the power of K is equal to some from zero to infinity. K n times X to the power of n Here's the up up psuedo facts is less than one part a. That gearbox is equal to the series k N X to end differentiate this series to show that the derivative of G well X is equal to K times GLX over one plus x here. Absolutely. If X is less than one, so behalf she x problem is equal to some from one to finally and times K and hams x two in my squad. Then we compute G of X prom hands one plus x, which is equal to some and from one to infinity. End times k n times x two m 11 as x two n And then we can rewrite this series as some and from zero to infinity 10 times K and us and plus one times Hey, Paswan hams extra m Then we compute and times k n plus plus one Homs a M plus one. So by definition, this is equal to 10 times. Hey, Hams K minus one times dot a dot times k minus And once one over in factorial us and as one harms Hey, times K minus one. I'm started all times K minus in over and plus one bacteria which is equal to K hams, came minus one. I'm started all times K minus in minus one. Offer a bacterial hams in plus a minus n which is a co two. Hey, hams. Mhm. So for half this is equal to Assam and from zero to infinity. Okay, Tom's K in hands X to end, which is equal to K Jail Fox. Yeah. So approved this equation Hard to be let h of X is equal to one plus X to the power of negative K video box and showed that HX prom is equal to zero. So we have HX crown is equal to negative K and spawn plus X the power of negative K minus one times to lux plus one plus x two negative K hams, the derivative of jacks. And if we use the result in part of a So this is hey hams. So you will g wax over one plus x so we can say this is a cultural zero sense this term. It's just K times one plus x to the powerful negative K minus one hams gearbox. Part of C deduced that gearbox is a go to one plus X to the power of K. So from part B, we have geo vax is equal to some constant number. C If we liked, X is equal to zero Behalf h zero it is c is equal to h zero is a cultural one times one which is a good one over half HX is 0 to 1, so Geo box is equal to one plus X to the power of K.