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10.4-bo'lim : Seriyalarning konvergentsiyasi/divergentsiyasi

1-4-masalalar uchun berilgan qatorlar uchun qisman yig‘indilar ketma-ketligida dastlabki 3 ta hadni hisoblang.

  1. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{1}{{1 + {3^n}}}} \)
  2. \( \displaystyle \sum\limits_{n = 1}^\infty {\left( {{2^n} - {3^n}} \right)} \)
  3. \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{1 + n}}{{2n}}} \)
  4. \( \displaystyle \sum\limits_{n = 0}^\infty {10} \)

5-7 muammolar uchun \(n\) deb faraz qilingth \( \displaystyle \sum\limits_{n = 0}^\infty {{a_n}} \) seriyasi uchun qisman yig'indilar ketma-ketligida atama quyida keltirilgan. \( \displaystyle \sum\limits_{n = 0}^\infty {{a_n}} \) seriyasining konvergent yoki divergentligini aniqlang. Agar qator konvergent bo'lsa, qatorning qiymatini aniqlang.

  1. \({s_n} = \left( {{n^2} + 4n} \right){{\bf{e}}^{ - 2n}}\)
  2. \(\displaystyle {s_n} = \frac{{1 + 2n + 3{n^2}}}{{4{n^2} + 5n + 6}}\)
  3. \(\displaystyle {s_n} = \frac{n}{{\ln \left( {n + 2} \right)}}\)
  4. \(\displaystyle {d_n} = \frac{{7 - 8n}}{{4 + 3n}}\) bo'lsin
    1. \(\left\{ {{d_n}} \right\}_{n = 0}^\infty \) ketma-ketligi yaqinlashadimi yoki ajraladimi?
    2. \( \displaystyle \sum\limits_{n = 0}^\infty {{d_n}} \) seriyasi yaqinlashadimi yoki ajraladimi?
  5. \({d_n} = 1 + n{{\bf{e}}^{ - n}}\) bo'lsin
    1. \(\left\{ {{d_n}} \right\}_{n = 0}^\infty \) ketma-ketligi yaqinlashadimi yoki ajraladimi?
    2. \( \displaystyle \sum\limits_{n = 0}^\infty {{d_n}} \) seriyasi yaqinlashadimi yoki ajraladimi?

10-12-masalalar seriyaning bir-biridan farqli ekanligini ko'rsatadi.

  1. \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{9 - 2{n^2}}}{{1 + 4n + {n^2}}}} \)
  2. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{{{5^n} + 1}}{{{3^n}}}} \)
  3. \( \displaystyle \sum\limits_{n = 1}^\infty {\cos \left( n \right)} \)