# Induction style proof

Bazman
Hi,

I need to prove the following:

$$1+ \frac{ 1}{ 2!} + \frac{1 }{3!} +...+ \frac{ 1}{ n!} < 2 \lbrack 1 - ( \frac{ 1}{2 } )^n \rbrack$$

From trying various example I'm fairly sure the relation holds but I can't seem to prove it algebraically?

Does the ineqaulity make a difference? Or can you behave pretty mcu as if it was an "=" ?

I tried simply doing 2[1-(1/2)^n] + 1/(n+1)! to try to get to 2[1-(1/2)^n+1]
but I can't seem to get very far?

Can anyone shed any light?

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murshid_islam
$$2 \left( 1 - \left( \frac{ 1}{2 } \right) ^n \right) = 1 + \frac{1}{2} + \frac{1}{2^2} + \cdots + \frac{1}{2^{n-1}}$$

we also know that $$2^n < n!$$

therefore,
$$\frac{1}{(n+1)!} < \frac{1}{2^{n+1}}$$

so,
$$2 \left( 1 - \left( \frac{ 1}{2 } \right) ^n \right) + \frac{1}{(n+1)!} < 2 \left( 1 - \left( \frac{ 1}{2 } \right) ^n \right) + \frac{1}{2^{n+1}}$$

$$2 \left( 1 - \left( \frac{ 1}{2 } \right) ^n \right) + \frac{1}{(n+1)!} < 2 \left( 1 - \left( \frac{ 1}{2 } \right) ^{n+1} \right)$$

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Bazman
thanks Murshid!

Its crystal clear now