10.18-bo'lim : binom seriyasi
3. \({\left( {1 + 3x} \right)^{ - 6}}\) uchun binom qatoridagi dastlabki to‘rtta hadni yozing.
Yechim
Birinchidan, biz bu holda biz \(k = - 6\) bilan to'g'ri shaklda bo'lgan eslatmalardan binom seriyasini ishlatish uchun to'g'ri shaklda ekanligiga ishonch hosil qilishimiz kerak.
2-qadam
Endi biz qilishimiz kerak bo'lgan yagona narsa eslatmalardan formulaga ulanish va dastlabki to'rtta shartni yozishdir.
\[\begin{align*}{\left( {1 + 3x} \right)^{ - 6}} & = \sum\limits_{i = 0}^\infty { {-6 \choose i} {{\left( {3x} \right)}^i}} \\ & = 1 + \left( { - 6} \right){\left( {3x} \right)^1} + \frac{{\left( { - 6} \right)\left( { - 7} \right)}}{{2!}}{\left( {3x} \right)^2} + \frac{{\left( { - 6} \right)\left( { - 7} \right)\left( { - 8} \right)}}{{3!}}{\left( {3x} \right)^3} + \cdots \\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{1 - 18x + 189{x^2} - 1512{x^3} + \cdots }}\end{align*}\]