Proving: 1+1/2!+1/3!...+1/n! < 2[1-(1/2)^n]

In summary, the conversation discusses proving the inequality 1 + (1/2!) + (1/3!) + ... + (1/n!) < 2[1 - (1/2)^n] algebraically. The use of various examples is mentioned and it is suggested to treat the inequality as an "=". The process of trying to prove the inequality is explained, including the use of the fact that 2^n < n! and the inequality 1/(n+1)! < 1/2^(n+1). The final step of the proof is shown, with the help of Murshid, to be 2[1 - (1/2)^n] + 1/(n+1)!
  • #1
Bazman
21
0
Hi,

I need to prove the following:

[tex] 1+ \frac{ 1}{ 2!} + \frac{1 }{3!} +...+ \frac{ 1}{ n!} < 2 \lbrack 1 - ( \frac{ 1}{2 } )^n \rbrack [/tex]

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?
 
Last edited:
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  • #2
[tex]2 \left( 1 - \left( \frac{ 1}{2 } \right) ^n \right) = 1 + \frac{1}{2} + \frac{1}{2^2} + \cdots + \frac{1}{2^{n-1}}[/tex]

we also know that [tex]2^n < n![/tex]

therefore,
[tex]\frac{1}{(n+1)!} < \frac{1}{2^{n+1}}[/tex]

so,
[tex]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}}[/tex]

[tex]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) [/tex]
 
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  • #3
thanks Murshid!

Its crystal clear now
 

What does the equation 1+1/2!+1/3!...+1/n! < 2[1-(1/2)^n] represent?

The equation represents a series of fractions where each term in the series is the reciprocal of the factorial of the corresponding number. The sum of these fractions is compared to 2 times the difference between 1 and 1 divided by 2 raised to the power of n.

Why is it important to prove this inequality?

This inequality is important because it has implications in various mathematical and scientific fields. It can be used to prove other mathematical theorems and it also has applications in fields such as probability, statistics, and engineering.

What is the significance of the term n! in the equation?

The term n! (n factorial) is a mathematical notation that represents the product of all positive integers from 1 to n. In this equation, it is used to show the relationship between the terms in the series and the exponent in the denominator of the second term.

How can this inequality be proven?

This inequality can be proven by using mathematical induction, where the base case is n = 1 and the inductive step is to show that if the inequality holds for n, it also holds for n+1. Another approach is to use the concept of convergence and prove that the series on the left side converges to a value less than 2 times the value on the right side.

What are some real-world applications of this inequality?

Some real-world applications of this inequality include its use in calculating probabilities in statistical analysis, in optimizing algorithms for computer science, and in designing experiments in engineering. It also has applications in finance, where it can be used to determine the expected value of certain investments.

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