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Homework Statement
If [tex]\overline{X}_n[/tex] converges to [tex]\mu[/tex], does [tex]\frac{1}{\overline{X}_n}[/tex] converge to [tex]\frac{1}{\mu}[/tex]?
Homework Equations
http://mathworld.wolfram.com/WeakLawofLargeNumbers.html
The Attempt at a Solution
[tex]\frac{1}{\overline{X}_n} = \frac{n}{X_1 + \cdots + X_n}[/tex]
[tex]E\left(\frac{1}{\overline{X}_n}\right) = E\left(\frac{n}{X_1 + \cdots + X_n}\right)[/tex]
[tex]E\left(\frac{1}{\overline{X}_n}\right) = n E\left(\frac{1}{X_1 + \cdots + X_n}\right)[/tex]
The RHS does not equal [tex]1/{\mu}[/tex] unless [tex]E\left(\frac{1}{X_1 + \cdots + X_n}\right) = \frac{1}{n \mu}[/tex] but how can I show that?
I know [tex]E\left( \frac{1}{\overline{X}_n} \right) \neq \frac{1}{\mu}[/tex] but I'm not sure how to apply that here.