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Topic: e=Napier's constant,.. e(x)=(1+1/x)**x
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Posts: 893
Registered: 12/6/04
e=Napier's constant,.. e(x)=(1+1/x)**x
Posted: Apr 24, 2005 4:49 AM
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Let e be Napier's constant and for x > , y>0 denote

e e
a(x,y) = ----------- , b(x,y)= --- ,
2(x+1)(y+1) 2xy

e(x)=(1+ 1/x)^{x} , E(x)=(1+ 1/x)^{x+1} .

Prove or disprove following :

E(y) - e(x)
a(x,y) < ----------- < b(x,y) for all x > 0,y > 0.

If O< x < y , then

E(y)- E(x) e(y)-e(x)
a(x,y) < --------- < b(x,y) , a(x,y) < --------- < b(x,y) .
x - y y - x

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