- #1
the_kid
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Homework Statement
I need to verify an integral representation of the Euler constant:
[itex]\int^{1}_{0}[/itex][itex]\frac{1-e^{-t}}{t}[/itex]dt-[itex]\int^{\infty}_{1}[/itex][[itex]\frac{e^{-t}}{t}[/itex]dt=[itex]\gamma[/itex]
Homework Equations
The Attempt at a Solution
OK, I'm supposed to use this fact (which I have already proved):
[itex]\sum^{N}_{n=1}[/itex][itex]\frac{1}{n}[/itex]=[itex]\int^{1}_{0}[/itex][itex]\frac{1-(1-t)^{n}}{t}[/itex]dt.
Then I am supposed to rescale t so that I can apply the follow definition of the exponential function:
lim as n-->infinity of (1+[itex]\frac{z}{n}[/itex])[itex]^{n}[/itex]=[itex]\sum^{\infty}_{k=0}[/itex][itex]\frac{z^{k}}{k!}[/itex]=e[itex]^{z}[/itex]
I'm not seeing how I can use the first fact at all...