Select the correct relation for $\left|{\int }_{\mathrm{C}} \mathrm{f}\left(\mathrm{z}\right) d\mathrm{z}\right|$ _____ ${\int }_{\mathrm{C}} \left|\mathrm{f}\left(\mathrm{z}\right)\right| \left|d\mathrm{z}\right|$ .

A sequence $\left\{\left(1-\frac{1}{{\mathrm{n}}^2}\right)+ \mathrm{i} \left(2+\frac{4}{\mathrm{n}}\right)\right\}$ converges to _____ .

The radius of convergence of power series $\sum _1^{\infty } {\left(1+\frac{1}{\mathrm{n}}\right)}^{{\mathrm{n}}^2} {\mathrm{z}}^{\mathrm{n}}$ is _____ .

The radius of convergence of power series $\underset{0}{\overset{\infty }{\sum }} \frac{2^{\mathrm{n}}}{\mathrm{n}!} {\mathrm{z}}^{\mathrm{n}}$ is _____ .

For which condition the series $\frac{1}{1-(z-1)}=1+(z-1)+{(z-1)}^2+.......$ is convergent?

Consider $\left|z\right|<1$ , then the Maclaurin series of $\log \left(1+\mathrm{z}\right)$ is _____ .

The Maclarin series of $\sin \mathrm{z}$ is _____ .

Let f(z) be analytic function in a simply connected domain D and let C be any closed continuous rectifiable curve in D, then

If f(z) is analytic function in the entire z-plane and bounded everywhere, then