Gottfried Leibniz: The Calculus Controversy n de 8.4 Problems Hint: Since her 1. Assuming Leibniz 's series za- 1 1 elet (1 +x)"=le logi+ du 4 3 5 full du 38 prove that = (1 + x)" log(1 +x), for 1 +...

Question 2

Extracted text: Gottfried Leibniz: The Calculus Controversy n de 8.4 Problems Hint: Since her 1. Assuming Leibniz 's series za- 1 1 elet (1 +x)"=le logi+ du 4 3 5 full du 38 prove that = (1 + x)" log(1 +x), for 1 + 9.11 JT the above limit is the value of the derivative the 8 5.7 (1 +x)" at u 0. (b) Next use the binomial series expansion of (1+x) to obtain mie 1 1 1 22-1 62-1 + 102- 1 + her s ones with 2. Given that P= n! denotes the number of permutations of n objects, establish the following identities 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-1 was ww 3! P2= P-1(P+1Pr). and eath. were rvive (1/n - 1)(1/n -2)(1/n - 3) ww 3. Verify Leibniz's famous identity, + 4! V6=1+-3+ 1--3, .She (i-) (c) From the fact lim -k, conclude born. il she n00 n which gives an imaginary decomposition of the real number 6. that itors, death itting er the on the 4. Obtain Mercator's logarithmic series log(1+x) x - .3 log(1+x) x 2 X + 3 4 4 7. Show that the binomial theorem, as stated by New in his letter to Oldenburg, is equivalent to the more for -1 < x="" 1,="" by="" first="" calculating="" by="" long="" division="" the="" series="" сcom-="" on="" the="" ion,="" a="" familiar="" form="" 1="" rir-="" 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="" time="" isible="" se="" the="" upiter="" lation,="" www.d="" 5.="" prove="" that="" 3!="" where="" r="" is="" an="" arbitrary="" integral="" or="" fractional="" expon="" the="" necessary="" condition="">< 1="" for="" convergence="" not="" stated="" by="" newton.="" log="" x="2" x="" +="" +="" 1="" -="" x="" 7="" for="" 1="">< x="">< 1, and hence 8. use the binomial theorem to obtain the following series expansions. comet log 2 = 2 it-on 1 (1+x)-l=1 -x+ x2 - r2 +.. + (-1)'x" +... curate 6. supply the details of the following derivation, due to euler, of the infinite series expansion for log(l +x): 3 5 35 3 33 either (a) (1 x)2=1+2r+ 3r2+.. +(n + 1)x" +. ... er than e same ly (a) (b) show t on be given by the limit + + + + + 1 -123 11 ii 1,="" and="" hence="" 8.="" use="" the="" binomial="" theorem="" to="" obtain="" the="" following="" series="" expansions.="" comet="" log="" 2="2" it-on="" 1="" (1+x)-l="1" -x+="" x2="" -="" r2="" +..="" +="" (-1)'x"="" +...="" curate="" 6.="" supply="" the="" details="" of="" the="" following="" derivation,="" due="" to="" euler,="" of="" the="" infinite="" series="" expansion="" for="" log(l="" +x):="" 3="" 5="" 35="" 3="" 33="" either="" (a)="" (1="" x)2="1+2r+" 3r2+..="" +(n="" +="" 1)x"="" +.="" ...="" er="" than="" e="" same="" ly="" (a)="" (b)="" show="" t="" on="" be="" given="" by="" the="" limit="" +="" +="" +="" +="" +="" 1="" -123="" 11="">