m21cfin2017key.pdf - Problem 1(10 points This problem...

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Problem 1(10 points).This problem concerns various definitions we havecovered in MAT 21C. Fill the blank with appropriate words or mathematicalexpressions.(1) An infinite sequencea1, a2, a3, . . .converges to a limitLif for everypositive number>0, there isN >0 so that|an-L|<holds for everynN. [n > Nis also fine.](2) An infinite seriesXn=1anconverges to a limitLif the partial sumSN=Nn=1anconverges toLasN→ ∞.(3) LetXn=1anxnbe a power series. A positive real number 0< R <is theradiusof convergenceof the series if it satisfies the following condition:If|x|< R, then the series is absolutely convergent, and if|x|> R, then the series is divergent.(4) Letf(x, y) be a function in two variables.The function valuecon-verges to a limitLas (x, y) approaches to an interior point (x0, y0)in the domain if for every positive>0, there is aδ >0 so that|f(x, y)-L|<holds for every (x, y) such thatp(x-x0)2+ (y-y0)2< δ.(5) Thepartial derivativeoff(x, y) with respect toxat a point (x0, y0)isfx(x0, y0) = limh0f(x0+h, y0)-f(x0, y0)h.Thepartial derivativeoff(x, y) with respect toyat a point (x0, y0)isfy(x0, y0) = limh0f(x0, y0+h)-f(x0, y0)h.1
2Problem 2(10 points total).This problem concerns the Taylor series ofarctan(x).One point for each answer, and work counts one point for (1), (2) and (4).(1) Find the Taylor series ofg(x) =1n=0Xn. Substi-
(2) Find the Taylor series of the integralRx0du1+u2atx= 0.
(3) Find the Taylor series of arctangent atx= 0.
(4) What is the radius of convergence of the above power series?
2. Since the geometric series of (1)has the radius of convergence 1 forX, the above series (3) is absolutelyconvergent for|x|<1, and divergent for|x|>1.
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