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GROUP 3IrwansyahAulia Khifah Futhona
Nurdianti RizkiHapsariNovita AtmasariFiryan Ramdhani
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Jakob and The Harmonic Series
Preliminaries...
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Theorem
In any finite geometric progression A, B,
C, . . . , D, E, the first term is to the secondas
the sum of all terms except the last is to thesum of all except the first.
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Let S = A + B + C + ... +
D + E
Proof:
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The Harmonic Series DIVERGES
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Lets check it out!
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JAKOB AND HIS FIGURATE SERIES
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Theorem N
d > 1, then
Proof
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Proof:
Hence
and so
heorem P :
If d > 1, x then
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Proof:Theorem
Cd > 1, then
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In a 1697 paper, general rule of Johann Bernoulli:
"The differential of a logarithm, no matter how
composed, is equal to the differential of the
expression divided by the expression
For instance, d[ln(x)]= orx
dx
( )[ ] ( )[ ]yyxxdyyxxd +=+ ln2
1ln
+
+
= yyxx
ydyxdx 22
2
1
yyxx
ydyxdx
+
+=
Johann and
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Johann described a somewhat complicated
geometric procedure for identifying the value of x
for which 1 + In x = 0
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Area under the curve from x = 0 to x = 12 Preliminaries
Substitute Nwithxx and zwith x ln x!
( ) ( ) ( )+
+
+= dxxx
m
nxx
mdxxx
nmnmnm 11ln
1ln
1
1ln
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The key to solving his curious problem.
Theorem:
( )
=
+=++=1
0 1
1
432
1
4
1
3
1
2
11
k
k
kx
kdxx
1
0
dxx xThe explanation is too long!!!
all terms in which are
found lx, orany power. . . of the natural
logarithm vanish, insofar as the
logarithm of unity is zero
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