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Confirmation: 1/2-1 +1 / 2 ^ 2-1 +...... 1 / 2 ^ n-1 <5 / 3

hylove842010-04-06 02:11:41 +0000 #1

shingkit2010-04-06 02:20:09 +0000 #2
Let T = 1/2-1 + 1 / 2 ^ 2-1 +...... 1 / 2 ^ n-1

T = 1 / 2 +1 / 2 ^ 2 +...... 1 / 2 ^ n - n

Let S = 1 / 2 +1 / 2 ^ 2 +...... 1 / 2 ^ n ............... (A)

T = S-n

Divide S by 2

1 / 2 S = 1 / 2 ^ 2 +...... 1 / 2 ^ n +1 ..................( B)

(A) - (B)

1 / 2 S = 1 / 2 - 1 / 2 ^ n +1

S = 1 - 1 / 2 ^ n

T = S - n = 1 - 1 / 2 ^ n-n

As n is an integer> 1,

T <0 <5 / 3

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