Calculus II PLTLFall 2014Worksheet 11These problems are to be do ne without the use of a calculator unless otherwisespecified.1) (Scribe) Let f(x) = x3+ x2+ x + 1.(a) Find the Taylor series of f at a = 1.(b) What expression will you get if you multiply o ut all of the terms o f the Taylorseries ? Explain.2) (Round Robin) Let f(x) = ln(1 − 2x) − tan−1(x/2).(a) Use the definition of a Taylor Series to find the coefficients of the terms of degreeless than five of the Taylor series at x = 0 for the function.(b) Use the series expression for ln(1 + x) and arctan(x) to check your answer.(c) What i s the interval of convergence for this series?3) (Pairs) Let f have a power series representation T , and suppose that(i) f(0) = 1(ii) f0(0) = 1(iii) f00(0) = 2(iv) f(3)(0) = 6(a) If the above is the only information we have a bout f, to what degree of accuracycan we estima te f(1)? Explain.(b) If, in addition, we know that T converges on the interval [−2, 2] and tha t|f(4)(x)| ≤ 15 on that interval, then to what degree of accuracy can we estim a te f(1)?4) (Round Robin) Find the radius of convergence and the i nterval of convergence of theseries .(a)P∞n=15nxn(b)P∞n=1xn/n5(c)P∞n=1nnxn(d)P∞n=1xn/nn5) (Scribe) The Maclaurin series for exis 1 + x +x22!+x33!+ · · ·, which is convergent forall x.(a) Give the Maclauri n series for 3exusing summation n o tation.(b) Give the Maclaurin s eries for e−3xusing summation notation.(c) Give the Maclaurin s eries for ex+ 3 using su mmation notation.(d) Give the Maclauri n series for ex+3using summation
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