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Does the following converge or diverge? $\int_{1}^{\infty} x^{-\frac{2}{3}} dx$

Does the following converge or diverge? $\int_{0}^{3} x^{-\frac{3}{2}} dx$

Does the following converge or diverge? $\int_{1}^{\infty} x^{-\frac{3}{2}} dx$

Prove that the real numbers are not countable.

Prove that if a function $f$ is bounded and monotonically decreasing on $[a,b]$ then it is Riemann-integrable on $[a,b]$.

Prove $f(x)$ is constant if $|f(x)-f(y)| < (x-y)^2$ for all real $x$ and $y$.

Find the limit $ \lim\limits_{x \to  0 } \frac{ \sin x – x \cos x}{ x^3 } $

Find the limit $ \lim\limits_{x \to  0 } \frac{ \tan x – x}{ x^3 } $

Will the following converge or diverge? $\sum\limits_{n=1}^\infty \frac{ (-1)^n x^n }{ \sqrt{1+n} } $

Will the following converge or diverge? $\sum\limits_{n=1}^\infty \frac{ 2^n x^n }{ n+1 } $