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Convergence generally means that a sequence of a certain sample quantity approaches a constant as the sample size tends to infinity. Convergence is also a property of an iterative algorithm to stabilize on some aim value.

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Does the normal distribution converge to a uniform distribution when the standard deviation ...

Your question is fundamentally flawed. The standard normal distribution is scaled so that $\sigma = 1$. So for some other Gaussian distribution ($\mu = 0, \sigma = \sigma^*$) then the curve between b …
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