#### MathematicalPhysicist

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if ##a_n>0## and ##\lim_{n\to \infty} a_{n+1}/a_n = L## then ##\lim_{n\to \infty} a_n^{1/n}=L##.

Now, the proof that I had seen for ##L\ne0## that we choose ##\epsilon<L##.

But what about the case of ##\epsilon>L##, in which case we have:

##a_{n+1}>(L-\epsilon)a_n## but the last RHS is negative, so I cannot take the n-th root without going into problems of an n-th root of a negative number, which is not defined for even n's in the real line.

I read this solution from Albert Blank's solutions to Fritz John and Richard Courant's textbook.