Gottfried Leibniz: The Calculus Controversynde8.4 ProblemsHint: Sinceher1. Assuming Leibniz 's seriesza-1 1elet(1 +x)"=le logi+du435fulldu38prove that= (1 + x)" log(1 +x),for1+9.11JTthe above limit is the value of the derivativethe85.7(1 +x)" at u 0.(b) Next use the binomial series expansion of(1+x) to obtainmie11122-1 62-1+102- 1+hersoneswith2. Given that P= n! denotes the number ofpermutations of n objects, establish the followingidentities from Leibniz 's Ars Combinatoria:(1/n- 1)log(1 +x) = lim x+2!ench(1/n- 1)(1/n-2)2 P,- (n - 1)P-1= P,+P-1wasww3!P2= P-1(P+1Pr).andeath.werervive(1/n - 1)(1/n -2)(1/n - 3)ww3. Verify Leibniz's famous identity,+4!V6=1+-3+ 1--3,.She(i-)(c) From the fact lim-k, concludeborn.il shen00nwhich gives an imaginary decomposition of the realnumber 6.thatitors,deathittinger theon the4. Obtain Mercator's logarithmic serieslog(1+x) x -.3log(1+x) x2X+3447. Show that the binomial theorem, as stated by Newin his letter to Oldenburg, is equivalent to the morefor -1 < x 1, by first calculating by long divisionthe seriesсcom-on theion, afamiliar form1rir- 1)= 1 -x+ x - x3+ .1 + x(1+x =1+rxr +2!and then integrating termwise between 0 and x.r(r - 1)- 2)+e timeisiblese theupiterlation,www.d5. Prove that3!where r is an arbitrary integral or fractional exponThe necessary condition x< 1 for convergencenot stated by Newton.logx= 2 x ++1 - x7for 1 < x < 1, and hence8. Use the binomial theorem to obtain the followingseries expansions.cometlog 2 = 2it-on1(1+x)-l=1 -x+ x2 - r2 +..+ (-1)'x" +...curate6. Supply the details of the following derivation, due toEuler, of the infinite series expansion for log(l +x):35 353 33either(a)(1 x)2=1+2r+ 3r2+..+(n + 1)x" +. ...er thane samely(a)(b)Show ton be given by the limit+++++1-12311II

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