A Pathological Function: Figure 12-8d shows the function Function f has derivatives of

Chapter 12, Problem 22

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A Pathological Function: Figure 12-8d shows the function Function f has derivatives of all orders atx = 0, and each derivative equals zero there.Figure 12-8da. By equating derivatives, show that theMaclaurin series for f(x) would be0 + 0x + 0x2 + 0x3 + .b. Show that the Maclaurin series converges forall values of x, but that it does not convergeto f(x) except at x = 0.c. Substitute x2 for x in the Maclaurin seriesfor ex. Write the first four terms of thepower series and simplify.d. The resulting power series is called aLaurent series, the name applied to a powerseries in which some powers can havenegative exponents. By finding a partial sumof the series, make a conjecture aboutwhether the Laurent series evaluated atx = 2 converges to f(2).