Problem 1546
What is the radius of convergence of the series \(\displaystyle \displaystyle\sum\limits^\infty_0 \frac{(n^2-n)!}{3^{n^2}}z^n\)? Difficulty: 3. |
Problem 1554
Find the radius of convergence of the following series. At which points do they converge, do they converge absolutely? What is their termwise derivative, antiderivative and what is the radius of convergence of those series? What is the largest disc with the same center as the power series to which these functions extend as regular functions? \(\displaystyle \sum_{n=0}^\infty z^n; \qquad \sum_{n=0}^\infty (n+1)(z+1)^n \qquad \sum_{n=0}^\infty \frac{(z-i)^n}{n!}; \qquad \sum_{n=1}^\infty \frac{(z+i)^n}n. \) Difficulty: 3. |
Problem 1547
Show that if \(\displaystyle f\) is the sum of a power series that converges on a disc of radius \(\displaystyle R\) around \(\displaystyle z_0\) then the average of \(\displaystyle f\) around a circle of radius \(\displaystyle r<R\) centered at \(\displaystyle z_0\) is \(\displaystyle f(z_0)\). Difficulty: 4. |
Problem 1548
For which \(\displaystyle z \in \C\) is \(\displaystyle \displaystyle\sum_{n=1}^\infty\frac{n^2}{3^n}(z+2i)^n\) convergent? Difficulty: 4. |
Problem 1549
For which \(\displaystyle z \in \C\) is \(\displaystyle \displaystyle\sum_{n=1}^\infty\frac{2^n}{3^n+5}(z+1-2i)^n\) convergent? Absolutely convergent? Difficulty: 4. |
Problem 1551
Find the Taylor series of \(\displaystyle 1/(z^2-1)\) around \(\displaystyle -2i\) and determine its radius of convergence. Difficulty: 4. |
Problem 1552
Find the Taylor series of \(\displaystyle 1/z\) around \(\displaystyle i\) and determine its radius of convergence. Difficulty: 4. |
Problem 1553
Find the Taylor series of \(\displaystyle 1/(z^2-1)\) around \(\displaystyle i\) and determine its radius of convergence. Difficulty: 4. |
Problem 1555
(a) \(\displaystyle \displaystyle f(z)=\sum\limits^\infty_0 \frac{z^n}{n}\) converges at all points on the unit circle except \(\displaystyle z=1\). (b) The function can be analytically continued along any of these points. Difficulty: 5. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |