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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Question 2

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 x< 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
help_outline

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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 x< 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

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Given that P = n! п 2P, - (п -1) Р,д — 2(п!) - (п-1)(п-1)) -2(п) - n(п-1) + (n-1)! (: п(п- 1)— т) 3 2(п!) - п+ (п-1)! — п!+ (п-1)! _ — Р, + Р., n- п

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