Question

8W (-1)+1 In this question, we will prove that - = In 2 a) Using the power series for centered at 0, express ln(1 - x) as a power series S(x) centered at 0. Show your work. b) Show that the interval of convergence for S(x) is (-1,1). Explain why this does not show that S(-1) = ln(2). c) We now wish to show that S(-1) = ln(2). Let P(x) be the Nth Taylor polynomial of ln(1 - x) at x = 0. Prove that for all ke N and x € (-1,0): P2x(x) < ln(1 – x) < P2x+1 (x) d) Conclude, by taking appropriate limits and being careful, that S(-1) = ln(2).

Answer 1

ANSWER:

Given that Nm lm2 a) The Power series to The Power Series for - centered at o = 14,24424434... lie 1 le ind14244224234...... Integrate above cuith respect to a we obtain - Ion(1-1)-9792423 + 2 My.. In(1-x)=-4-12 I b) Here Sta) - 3 gan for skal anse", we have an- * * Meln. for interval of convergence alim la comuniki laki (ov) stad</ for tal, slu) = divergences <x< 1

708 X:-1, S(u):- (-1)". § (-1)" N=1 " nañ convergence Thus, imbesval of convergence is tsa21 of 9€ (-11) c) sco): ce con 2 Main Also Rap () and Przelew Papcale Incl-a)< Part1 (2) d) St1):-42 +93 n=1 S(-1) = |(2).