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15h
comment Peculiar family of apparently positive semidefinite matrices
See Achilles Hui's answer here math.stackexchange.com/questions/577197/…
19h
comment Show that $(1+\frac{1}{n})^n+\frac{1}{n}$ is eventually increasing
Also note that the terms of the form $1 - \dfrac{k}{n+1}$ are less than 1 in absolute value.
19h
comment Show that $(1+\frac{1}{n})^n+\frac{1}{n}$ is eventually increasing
Use the Binomial theorem, the $k^{th}$ term is \begin{align*} (-1)^k {n+1 \choose k} \frac{1}{(n+1)^{2k}} &=\frac{(-1)^k}{k!} \frac{1}{(n+1)^k} \frac{ (n+1)(n+1 - 1) \dots (n+1 - (k-1))}{(n+1)(n+1) \dots (n+1)} \\ &= \frac{(-1)^k}{k!} \dfrac{1}{(n+1)^k} \left(1 - \frac{1}{n+1}\right) \dots \left(1 - \frac{k-1}{n+1}\right). \end{align*} The term you are interested comes from setting $k = 4$
22h
answered Show that $(1+\frac{1}{n})^n+\frac{1}{n}$ is eventually increasing
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revised Prove that the limit exists and then find its limit
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answered Prove that the limit exists and then find its limit
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comment real analysis on function continuity and rational numbers
Every subset of $Z$ is open. If you consider $Z$ as a metric space with $d(x,y)=|x-y|$ then $\{1\} = \{x : d(x,1) < 1/2 \}$.
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answered real analysis on function continuity and rational numbers
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