10.4-bo'lim : Seriyalarning konvergentsiyasi/divergentsiyasi
2. Quyidagi qatorlar uchun qisman yig‘indilar ketma-ketligidagi dastlabki 3 ta hadni hisoblang.
\[\sum\limits_{n = 3}^\infty {\frac{{2n}}{{n + 2}}} \]Yechim
\(n\) ekanligini unutmangth qisman yig'indilar ketma-ketligidagi atama qatorning birinchi \(n\) shartlari yig'indisidir. Shunday qilib, dastlabki uchta shartni qisman yig'indilar ketma-ketligida hisoblash juda oddiy.
Mana bu muammo uchun ish.
\[\require{bbox} \bbox[2pt,border:1px solid black]{\begin{align*}{s_3} & = \frac{{2\left( 3 \right)}}{{3 + 2}} = \frac{6}{5}\\ {s_4} &= \frac{{2\left( 3 \right)}}{{3 + 2}} + \frac{{2\left( 4 \right)}}{{4 + 2}} = \frac{{38}}{{15}}\\ {s_5} & = \frac{{2\left( 3 \right)}}{{3 + 2}} + \frac{{2\left( 4 \right)}}{{4 + 2}} + \frac{{2\left( 5 \right)}}{{5 + 2}} = \frac{{416}}{{105}}\end{align*}}\]