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Compute $\limsup |\sin^n (n)|$

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#real-analysis #limsup-and-liminf
128
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Prove $\lim_{n\to\infty} (x_n^{1/k})$ = $(\lim_{n\to\infty} x_n)^{1/k}$

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#real-analysis #sequences-and-series #proof-verification
279
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rate of convergence of two sequences' product

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#real-analysis
120
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$\int\frac1{(1+x^2)^3}\,dx$ without Hermite

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#real-analysis #polynomials #indefinite-integrals #hermite-polynomials
214
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Does $\sum_{n=1}^{\infty}(\sin{n})^n$ converge?

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#real-analysis
111
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Show that the only curves whose curvature is fixed constant $k$ are circles of radius $1/k$.

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#real-analysis #differential-geometry
63
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If f:X→R is a continuous non constant function and X is connected then X is uncountable.

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#real-analysis #metric-spaces
237
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Euler-Lagrange equations without lower boundary conditions

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#real-analysis #functional-analysis #calculus-of-variations #boundary-value-problem #euler-lagrange-equation
61
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Prove that $u(x)-u(y) \le d(x,y) \le u(x)+u(y)$

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#calculus #real-analysis #metric-spaces
62
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Prove that $\lim_{n\to \infty} \int_{a}^{b}f(nx)dx=\frac{b-a}{T}\int_{0}^{T}f(t)dt.$

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#real-analysis #integration
39
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Convolution of $\delta$ and $f$, in $L^1$ space

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#real-analysis #convolution
133
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Prove or disprove differentiability

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#real-analysis #derivatives
116
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Show $\sum_\limits{n=1}^{\infty}(1+\frac{1}{n})^{n^2}z^n$ is convergent.

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#calculus #real-analysis #sequences-and-series #complex-analysis
38
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Convergence of an Integral iff another Integral Converges

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#real-analysis #integration #convergence-divergence
193
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Show that $\lim_\limits{n\to \infty} \int_{a}^{b}f(x)\cos(nx)dx=0$

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#real-analysis #integration #estimation

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