Problem 1164
By the binomial theorem \(\displaystyle \displaystyle (1+x)^\alpha = \sum_{k=0}^\infty \binom{\alpha}{k} x^k\) if \(\displaystyle |x|<1\). Which identities result in the \(\displaystyle \alpha=-1\) and \(\displaystyle \alpha=-2\) cases? Difficulty: 1. |
Problem 1172
For which \(\displaystyle c\in \R\) \(\displaystyle \sum_{k=0}^\infty \binom{c}{k} = 2^c ? \) Difficulty: 2. |
Problem 1163
Determine the radius of convergence of the following series. \(\displaystyle \sum n^{99}x^n \qquad \sum\left(1+\frac1n\right)^{\!n^2}\!x^n \quad \sum n!x^{n^2} \) Difficulty: 3. |
Problem 1161
Determine the Taylor series of the function at the given point. (a) \(\displaystyle \displaystyle\frac1{1-x}\) at \(\displaystyle 0\); (b) \(\displaystyle \displaystyle\frac1{x^2}\) at \(\displaystyle 3\); (c) \(\displaystyle \log x\) at \(\displaystyle 5\) körül; (d) \(\displaystyle \sin x\) at \(\displaystyle \displaystyle\frac{\pi}3\); (e) \(\displaystyle \log(x^2-1)\) at \(\displaystyle 2\); (f) \(\displaystyle \displaystyle\arsh x^2\) at \(\displaystyle 0\); (g) \(\displaystyle \arcth x\) at \(\displaystyle 2\). Give intervals where the Taylor series converges to the function. Difficulty: 4. |
Problem 1169
Determine the Taylor series of \(\displaystyle \arth x\) around \(\displaystyle a=1/2\). For which \(\displaystyle x\) do the series equal the original function? Difficulty: 5. |
Problem 1171
(a) For which real values of \(\displaystyle c\) will the series \(\displaystyle \displaystyle\sum_{n=1}^\infty\Big(n^c\cdot\cos(nx)\Big)\) converge on \(\displaystyle \R\)? (b) For which real values of \(\displaystyle c\) will the series \(\displaystyle \displaystyle\sum_{n=1}^\infty\Big(n^c\cdot\sin(nx)\Big)\) converge uniformly on \(\displaystyle \R\)? Difficulty: 5. |
Problem 1165
\(\displaystyle \frac{x}1 - \frac{x^3}3 + \frac{x^5}5 - \frac{x^7}7 +- \ldots = ? \qquad \sum_{k=0}^\infty\frac{(-1)^k}{4k+1} = ? \) Difficulty: 6. |
Problem 1166
\(\displaystyle \sum_{k=0}^\infty\left(\frac1{3k+1}-\frac1{3k+2}\right) = ? \) Difficulty: 6. |
Problem 1167
Let \(\displaystyle c_0=1\) and \(\displaystyle c_{n+1}=\sum\limits_{k=0}^nc_kc_{n-k}\). (Catalan numbers.) Define \(\displaystyle G(x)=\sum\limits_{n=0}^\infty c_nx^n\) the so called generating function of the Catalan-numbers. (a) Prove that \(\displaystyle G\) converges in a neigborhood of \(\displaystyle 0\). (b) Prove that in the (non-empty) interior of the convergence interval \(\displaystyle G(x)=xG^2(x)+1\). (c) Using b) determine \(\displaystyle G\) and \(\displaystyle c_n\) explicitely. Difficulty: 6. |
Problem 1170
\(\displaystyle \sum_{k=0}^\infty \left(\frac1{3k+1}+\frac1{3k+2}-\frac2{3k+3}\right) = ? \) Difficulty: 6. |
Problem 1162
Construct an infinitely differentiable function \(\displaystyle f\) whose Taylor series around \(\displaystyle 0\) converges everywhere but the limit equals \(\displaystyle f(x)\) if and only if \(\displaystyle x\in [-1,1]\). Difficulty: 7. |
Problem 1168
Let \(\displaystyle p_n\) be the number of partitions of the number \(\displaystyle n\) into different parts. For example \(\displaystyle p_0=1\) and \(\displaystyle p_6=4\), because \(\displaystyle 6=5+1=4+2=3+2+1\).) Using the generating series \(\displaystyle P(x)=\sum\limits_{n=0}^\infty p_nx^n\) find an upper bound for \(\displaystyle p_n\). Difficulty: 8. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |