Integral of Reciprocal is Divergent
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Theorem
Unbounded Above
- $\ds \int_1^n \frac {\d x} x \to +\infty$ as $n \to + \infty$
To Zero
- $\ds \int_\gamma^1 \frac {\d x} x \to -\infty$ as $\gamma \to 0^+$
Thus the improper integrals $\ds \int_1^{\to +\infty} \frac {\d x} x$ and $\ds \int_{\to 0^+}^1 \frac {\d x} x$ do not exist.
In particular:
$\ds \int_{\to 0^+}^{\to +\infty} \frac {\d x} x$
certainly does not exist.
Also see
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 13.33$