Arnold F. answered • 11/21/15
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Have you tried proof by induction:
First prove statement (formula) is true for n=1
Next assume true for n=k
Then prove true for n= k +1
Arnold F.
Since the subscript of the formula is n-1 you really use n=2 in the formula to calculate S1. It is a little misleading that the subscript in the series is using n and the subscript in the formula is n-1. They should have been different letters.
So the first formula would be: 1/√1 = S1 = S2-1 > 2(√2 -1). This is the basis step.
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11/21/15
Muaaz K.
I have reached the final step:
n= k + 1 where I have the line:
Sk-2 > 2((k-1)1/2 - 1)
from this point how would I go about proving this? I was thinking of splitting the RHS into (Sk-1)-1?
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11/21/15
Arnold F.
Since we are assuming Sk > 2(√(k+1) - 1) is true
we want to show Sk+1 > 2(√(k+2) -1) (2)
Sk+1 = Sk + 1/√(k+1)
> 2((√(k+1) - 1) + 1/√(k+1)
Manipulate the right side of the last inequality until it looks like (2)
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11/21/15
Muaaz K.
For your assumption are you setting n = k+1?
and then proving n = k+2?
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11/22/15
Arnold F.
The short answer is yes. In a proof by induction, once you assume a value of the subscript is true it is sufficient to then prove the next subscript is true.
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11/22/15
Muaaz K.
11/21/15