1. Oblicz sumy częściowe \(s_n=\displaystyle\sum_{k=1}^n a_k\), a następnie znajdź \(\displaystyle\lim_{n\to\infty}s_n\):
    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. \(\displaystyle\sum_{n=1}^\infty\big(\sqrt{n+2}-\sqrt{n}\big)(-1)^n\).