10.4-bo'lim: Seriyalarning konvergentsiyasi/divergentsiyasi

1 va 2 masalalar uchun berilgan qatorlar uchun qisman yig‘indilar ketma-ketligida dastlabki 3 ta hadni hisoblang.

1. \( \displaystyle \sum\limits_{n = 1}^\infty {n\,{2^n}} \) Yechim
2. \( \displaystyle \sum\limits_{n = 3}^\infty {\frac{{2n}}{{n + 2}}} \) Yechim

3 va 4 muammolar uchun \(n\) deb faraz qiling^th \( \displaystyle \sum\limits_{n = 0}^\infty {{a_n}} \) 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. \(\displaystyle {s_n} = \frac{{5 + 8{n^2}}}{{2 - 7{n^2}}}\) Yechim
2. \(\displaystyle {s_n} = \frac{{{n^2}}}{{5 + 2n}}\) Yechim

5 va 6 muammolar uchun ketma-ketlik farqli ekanligini ko'rsatadi.

1. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{{3n\,{{\bf{e}}^n}}}{{{n^2} + 1}}} \) Yechim
2. \( \displaystyle \sum\limits_{n = 5}^\infty {\frac{{6 + 8n + 9{n^2}}}{{3 + 2n + {n^2}}}} \) Yechim