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Define $L(x)=\int_{1}^x {1\over t} dt $

NOTE: I realize that $L(x)$ is the definition of $ln(x)$, but we aren't allowed to use that. Our professor is walking us through the definition of $ln(x)$ and $e^x$.

Part A: Show that $L({1\over x}) = -L(x)$

I've tried several different substitutions for this and even direct proof, but I'm completely stuck after working/thinking about this for the past several hours. Help is greatly appreciated!

Part B: Using Cauchy Criterion showing that the sequence $$s_n = 1 + {1 \over 2} + {1 \over 3} + ... + {1 \over n}$$ is divergent if $m>n$, show that $L(x)$ tendsto $\infty$ as $x \to \infty$.

Basically, I'm trying to show that if the limit of the sequence converges to some L, then the function of that sequence also converges to the same L. I suspect that I need to show that $l(x)=s_n={1 \over x}$, but I don't know how to formally state this idea.

  • 2
    A. When you write the definition of $L(1/x)$, what do you get? Is there a simple change of variables you can use to relate the new interval of integration to the interval $[1,x]$?2012-12-12
  • 0
    Is Cauchy criterion the fact that convergent sequences are Cauchy, or that the convergence of the series for a monotonic sequence is determined by the $2^n$th terms, more usually called Cauchy condensation test?2012-12-12

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