1. Znajdź promień zbieżności szeregów potęgowych:
    1. \(\displaystyle\sum_{n=1}^\infty\frac{\binom{3n}{n}\,x^n}{n^2}\),
    2. \(\displaystyle\sum_{n=1}^\infty\frac{2^{n+7}x^{6n}}{\sqrt{n}}\),
    3. \(\displaystyle\sum_{n=1}^\infty\frac{(54n+1)^nx^{3n}}{(81n+2)^n}\),
    4. \(\displaystyle\sum_{n=1}^\infty10^{n^2}x^{n^3}\),
    5. \(\displaystyle\sum_{n=1}^\infty n!\,x^{2^n}\),
    6. \(\displaystyle\sum_{n=1}^\infty\frac{10^nx^n}{n^{10}}\),
    7. \(\displaystyle\sum_{n=1}^\infty\frac{x^n}{n\,10^{n-1}}\),
    8. \(\displaystyle\sum_{n=1}^\infty50^n\,x^{2n+5}\),
    9. \(\displaystyle\sum_{n=1}^\infty\frac{x^n}{n(n+1)}\),
    10. \(\displaystyle\sum_{n=1}^\infty\frac{x^{2n}}{\sqrt{n^2+n}-n}\),
    11. \(\displaystyle\sum_{n=1}^\infty\frac{4^{n+5}x^{3n+7}}{n\,6^{2n}}\),
    12. \(\displaystyle\sum_{n=1}^\infty\frac{(2n)!x^n}{(n!)^3}\),
    13. \(\displaystyle\sum_{n=1}^\infty\frac{n!}{n^n}x^{n+7}\),
    14. Niespodzianka: nie ma punktu n.
    15. \(\displaystyle\sum_{n=1}^\infty\binom{4n}{n}x^n\),
    16. \(\displaystyle\sum_{n=1}^\infty n!\,x^{n^2}\),
    17. \(\displaystyle\sum_{n=1}^\infty\binom{n+10}{n}x^n\),
    18. \(\displaystyle\sum_{n=1}^\infty\frac{n!\,(3n)!}{(2n)!\,(2n)!}\,x^n\).
  2. Znajdź granice:
    1. \(\displaystyle\lim_{x\to7}\Big(\frac{1}{x-7}-\frac{8}{x^2-6x-7}\Big)\),
    2. \(\displaystyle\lim_{x\to0}x\sin\Big(\frac{1}{x}\Big)\),
    3. \(\displaystyle\lim_{x\to4}\frac{\sqrt{x}-2}{x-4}\),
    4. \(\displaystyle\lim_{x\to3}\frac{x-3}{x+2}\),
    5. \(\displaystyle\lim_{x\to5}\frac{x^2-6x+5}{x-5}\),
    6. \(\displaystyle\lim_{x\to1}\Big(\frac{1}{1-x}-\frac{3}{1-x^3}\Big)\),
    7. \(\displaystyle\lim_{x\to1}\frac{x^{2007}-1}{x^{10}-1}\),
    8. \(\displaystyle\lim_{x\to1/2}\frac{8x^3-1}{6x^2-5x+1}\),
    9. \(\displaystyle\lim_{x\to-2}\frac{x^3+3x^2+2x}{x^2-x-6}\),
    10. \(\displaystyle\lim_{x\to0^+}\frac{x-\sqrt{x}}{\sqrt{x}}\),
    11. \(\displaystyle\lim_{x\to1}\frac{(x-1)\sqrt{2-x}}{x^2-1}\).