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Limit of $∞.0$ form of an integral and Riemann sum

if $f\in C^2[0,1]$,we have the conclusion: $\begin{aligned}\lim_{n\rightarrow\infty}n\left(\frac{1}{n}\sum_{k=0}^nf\left(\frac{k}{n}\right)-\int_0^1f(x)\mathrm{d}x\right)=\frac{1}{2}[f(1)-f(0)]\end{...
RainField's user avatar
1 vote

Compute the limit of the Log-Sum-Exp function

Here's a hint if you want to try it yourself first: what happens if you divide by $\exp(\rho x_j)$ where $x_j$ is a maximal element? Choose $x_j$ so that $x_j = \max_i (x_1, \ldots, x_N)$. We then ...
hff1's user avatar
  • 168
1 vote

Let $a_n$ be the sequence defined inductively by $a_1 = 2$ and $a_{n+1} =(1/2)(a_n+2/a_n)$. Prove that $(a_n)^2 ⩾ 2$ for $n ∈ N$.

Proof via induction. Basis case: $a_1 > \sqrt{2}.$ Induction case: Using AM-GM, $a_n > 2 \implies a_{n+1} = \frac{a_n + \frac{2}{a_n}}{2} > \sqrt{(a_n)\left( \frac{2}{a_n} \right)} = \sqrt{2}....
Adam Rubinson's user avatar

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