- #1
librastar
- 15
- 0
Homework Statement
Show that Euler's constant is 0 < [tex]\gamma[/tex] < 1
Homework Equations
According to my book, [tex]\gamma[/tex] = lim((1+1/2+...+1/n) - log n) as n approaches infinity
The Attempt at a Solution
At first glance I was thinking about proving by contradiction.
First I assume [tex]\gamma[/tex] is equal to 0.
Then I will get lim(1+1/2+...+1/n) = lim (log n) as n approaches infinity.
However 1+1/2+...+1/n is a harmonic series which diverges, so the equality does not hold.
So [tex]\gamma[/tex] does not equal 0.
Next I assume [tex]\gamma[/tex] is less than 0.
Since log function never yields negative result, this implies that lim(1+1/2+...+1/n) < lim(log n) as n approaches infinity.
Again due to the divergent nature of harmonic series, the inequality does not hold.
These are all I have for now.
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I feel I have logic error in the attempt to solve the problem, and I hope that someone will give me a hint on the correct solution.