Does this limit help answer whether the gamma constant is irrational or transcendent?

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Let γ=Limit[Sum[k((k*t+(k-1))*(t+1)^1/k-(k*(t+1))*t^1/k)/((k-1)*t*(t+1)),{t,1,∞}],k->1], is it easier to see if γ∈ℚ?

 
For those of us who are fluent in various math areas but not this one (whatever it is), could you please explain what the problem is?

E.g., what is the gamma constant, and perhaps can your very dense limit expression be broken down a bit?

Thanks.
 

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