Let a1=0, a2n+1=a2n=n−an. Prove that there exists k such that |ak−k3|>2010 and yet lim.
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Let a1=0, a2n+1=a2n=n−an. Prove that there exists k such that |ak−k3|>2010 and yet lim.
Let f be a differentiable function. Prove that if \lim_{x\to\infty} (f(x)+f'(x))=1, then \lim_{x\to\infty} f(x)=1.
Evaluate the following limit:
\displaystyle \lim_{\varepsilon\to 0}\int_0^{2\varepsilon} \log\left(\frac{|\sin t-\varepsilon|}{\sin \varepsilon}\right) \frac{dt}{\sin t}.
Let a_1<\cdots be a sequence of positive integers such that \log a_1, \log a_2,\log a_3,\cdots are linearly independent over the rational field \mathbb Q. Prove that \lim_{k\to \infty} a_k/k=\infty.
Let a_1=\sqrt{1+2},
a_2=\sqrt{1+2\sqrt{1+3}},
a_3=\sqrt{1+2\sqrt{1+3\sqrt{1+4}}}, …,
a_n=\sqrt{1+2\sqrt{1+3\sqrt {\cdots \sqrt{\sqrt{\sqrt{\cdots\sqrt{1+n\sqrt{1+(n+1)}}}}}}}}, … .
Prove that \displaystyle\lim_{n\to \infty} \frac{a_{n+1}-a_{n}}{a_n-a_{n-1}}=\frac12.