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  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}.