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  1. Oblicz sumy częściowe sn=n∑k=1ak, a następnie znajdź lim:
    1. a_k=\displaystyle\frac{1}{5^k},
    2. a_k=\displaystyle\frac{2^k+5^k}{10^k}.
  2. Udowodnij, że szereg \displaystyle\sum_{n=1}^\infty\frac{1}{2^n-1} jest zbieżny, a jego suma jest mniejsza od 2.
  3. Rozstrzygnij, czy następujące szeregi są zbieżne (k!! oznacza iloczyn wszystkich liczb naturalnych nie większych od k o tej samej parzystości):
    1. \displaystyle\sum_{n=1}^\infty\frac{1}{n^2+1},
    2. \displaystyle\sum_{n=2}^\infty\frac{1}{n^2-1},
    3. \displaystyle\sum_{n=1}^\infty\frac{1+n}{n^2+1},
    4. \displaystyle\sum_{n=1}^\infty\frac{2\cdot 5\cdot 8\cdot\dots\cdot(3n-1)}{1\cdot 5\cdot 9\cdot\dots\cdot(4n-3)},
    5. \displaystyle\sum_{n=1}^\infty\frac{5n^2-1}{n^3+6n^2+8n+47},
    6. \displaystyle\sum_{n=1}^\infty \frac{1}{(2n-1)\cdot2^{2n-1}},
    7. \displaystyle\sum_{n=1}^\infty\frac{1}{3n-1},
    8. \displaystyle\sum_{n=1}^\infty\frac{1}{\sqrt{n^2+2n}},
    9. \displaystyle\sum_{n=1}^\infty \frac{1}{(n+1)(n+4)},
    10. \displaystyle\sum_{n=1}^\infty\frac{1}{(2n+1)!},
    11. \displaystyle\sum_{n=1}^\infty\frac{n^2}{3^n},
    12. \displaystyle\sum_{n=1}^\infty\frac{(2n-1)!!}{3^nn!},
    13. \displaystyle\sum_{n=1}^\infty\Big(\frac{n}{2n+1}\Big)^n,
    14. \displaystyle\sum_{n=1}^\infty\frac{\left(\frac{n+1}{n}\right)^{n^3}}{3^n},
    15. \displaystyle\sum_{n=2}^\infty\frac{1}{(n-1)\sqrt{n+1}},
    16. \displaystyle\sum_{n=1}^\infty\sqrt{\frac{n+1}{n}},
    17. \displaystyle\sum_{n=1}^\infty\frac{n^2}{n!},
    18. \displaystyle\sum_{n=1}^\infty\frac{n}{2n-1},
    19. \displaystyle\sum_{n=1}^\infty\frac{2^n}{n^4},
    20. \displaystyle\sum_{n=1}^\infty\frac{1}{\sqrt{n^2+n}-n},
    21. \displaystyle\sum_{n=1}^\infty\frac{1000^n}{\root 10\of{n!}},
    22. \displaystyle\sum_{n=1}^\infty\frac{\arctan n}{n^2+\arctan n},
    23. \displaystyle\sum_{n=1}^\infty\frac{3^n}{2^{2^n}},
    24. \displaystyle\sum_{n=1}^\infty\frac{n^3+\pi}{n^\pi +e}.
  4. Które z następujących szeregów są zbieżne, a które są zbieżne absolutnie:

    1. \displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}}{2n-1},

    2. \displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^23^n},

    3. \displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}}{(2n-1)^3},

    4. \displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}n+1}{n},

    5. \displaystyle\sum_{n=1}^\infty\frac{1}{\sqrt{(n+4)(n+9)}},

    6. \displaystyle\sum_{n=1}^\infty\frac{(-1)^n\cdot2^{10^n}}{3^{2^n}},

    7. \displaystyle\sum_{n=1}^\infty\frac{n!\cdot(-5)^n}{n^n\cdot2^n},

    8. \displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}n^3}{2^n},

    9. 1-1+1-\frac12-\frac12+1-\frac13-\frac13-\frac13+\dots+1-\stackrel{k\text{ razy}}{\overbrace{\frac1k-\frac1k-\dots-\frac1k}}+\dots,

    10. 1-1+\frac12-\frac14-\frac14+\frac13-\frac19-\frac19-\frac19+\dots+\frac1k-\stackrel{k\text{razy}}{\overbrace{\frac1{k^2}-\frac1{k^2}-\dots-\frac1{k^2}}}+\dots,

    11. \displaystyle\sum_{n=2}^\infty\frac{(-1)^n}{n-\sqrt{n}},

    12. \displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}2^{n^2}}{n!},

    13. \displaystyle\sum_{n=1}^\infty\frac{\sin 77n}{n^2},

    14. \displaystyle\sum_{n=1}^\infty\frac{2^n+17}{3^n},

    15. \displaystyle\sum_{n=1}^\infty\frac{\sqrt{n!+1}}{n!},

    16. \displaystyle\sum_{n=1}^\infty\frac{(-1)^{n^2}}{(n+3)^{1/4}},

    17. \displaystyle\sum_{n=1}^\infty\frac{n+2}{n(n+1)}(-1)^n,

    18. \displaystyle\sum_{n=1}^\infty\frac{(-1)^n}{\sqrt{n}} \left(1+\frac{(-1)^n}{\sqrt{n}}\right),

    19. \displaystyle\sum_{n=1}^\infty\frac{2^n}{n\sqrt{4^n+3^n}},

    20. \displaystyle\sum_{n=1}^\infty\frac{1}{n+5\sqrt{n}+27},

    21. \displaystyle\sum_{n=1}^\infty\frac{\binom{2n}{n}}{n!},

    22. \displaystyle\sum_{n=1}^\infty\frac{2^{n^2}}{4^{\binom{n}{2}}},

    23. \displaystyle\sum_{n=1}^\infty\frac{(-1)^n}{n^{1/n}},

    24. \displaystyle\sum_{n=1}^\infty\frac{(\frac{n+1}{n})^{n^2}}{2^n},

    25. \displaystyle\sum_{n=1}^\infty\frac{(-1)^n(\frac{n+1}{n})^{n^2}}{3^n},

    26. \displaystyle\sum_{n=3}^\infty\frac{(\log n)^{\log n}(-1)^n}{n^{\log\log n}},

      ż. \displaystyle\sum_{n=1}^\infty\frac{(-1)^n}{\arctan n},
      ź. \displaystyle\sum_{n=1}^\infty\big(\sqrt{n+2}-\sqrt{n}\big)(-1)^n.