10.14-bo'lim : Quvvat seriyasi
4. Quyidagi darajalar qatori uchun yaqinlashish oraliq va radiusini aniqlang.
\[\sum\limits_{n = 0}^\infty {\frac{{{4^{1 + 2n}}}}{{{5^{n + 1}}}}{{\left( {x + 3} \right)}^n}} \]Yechim
Mayli, keling, \(L\) bilan tanishish uchun nisbat testidan boshlaylik.
\[L = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{{4^{3 + 2n}}{{\left( {x + 3} \right)}^{n + 1}}}}{{{5^{n + 2}}}}\frac{{{5^{n + 1}}}}{{{4^{1 + 2n}}{{\left( {x + 3} \right)}^n}}}} \right| = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{{4^2}\left( {x + 3} \right)}}{5}} \right| = \left| {x + 3} \right|\mathop {\lim }\limits_{n \to \infty } \frac{{16}}{5} = \frac{{16}}{5}\left| {x + 3} \right|\]2-qadam
Shunday qilib, biz bilamizki, seriyalar birlashadi, agar:
\[\frac{{16}}{5}\left| {x + 3} \right| < 1\hspace{0.25in} \to \hspace{0.25in}\left| {x + 3} \right| < \frac{5}{{16}}\]3-qadam
Shunday qilib, oldingi bosqichdan biz yaqinlashuv radiusi \(\require{bbox} \bbox[2pt,border:1px solid black]{{R = \frac{5}{{16}}}}\) ekanligini ko'ramiz.
4-qadam
Keling, konvergentsiya oralig'i ustida ishlashni boshlaylik. Keling, 2-bosqichda mavjud bo'lgan tengsizlikni yo'q qilaylik.
\[ - \frac{5}{{16}} < x + 3 < \frac{5}{{16}}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in} \to \hspace{0.25in}\hspace{0.25in} - \frac{{53}}{{16}} < x < - \frac{{43}}{{16}}\]5-qadam
Konvergentsiya oralig'ini yakunlash uchun biz 4-bosqichdagi tengsizlikning oxirgi nuqtalarini tekshirishimiz kerak.
\( \displaystyle x = - \frac{{53}}{{16}}\,\,\,:\,\,\,\sum\limits_{n = 0}^\infty {\frac{{{4^1}{4^{2n}}}}{{{5^n}{5^1}}}{{\left( { - \frac{5}{{16}}} \right)}^n}} = \sum\limits_{n = 0}^\infty {\frac{{4\left( {{{16}^n}} \right)}}{{{5^n}\left( 5 \right)}}\frac{{{{\left( { - 1} \right)}^n}{5^n}}}{{{{16}^n}}}} = \sum\limits_{n = 0}^\infty {\frac{{4{{\left( { - 1} \right)}^n}}}{5}} \)\(\displaystyle x = - \frac{{43}}{{16}}\,\,\,:\,\,\,\sum\limits_{n = 0}^\infty {\frac{{{4^1}{4^{2n}}}}{{{5^n}{5^1}}}{{\left( {\frac{5}{{16}}} \right)}^n}} = \sum\limits_{n = 0}^\infty {\frac{{4\left( {{{16}^n}} \right)}}{{{5^n}\left( 5 \right)}}\frac{{{5^n}}}{{{{16}^n}}}} = \sum\limits_{n = 0}^\infty {\frac{4}{5}} \)
Endi,
\[\mathop {\lim }\limits_{n \to \infty } \frac{{4{{\left( { - 1} \right)}^n}}}{5}\,\,\,\,\, - \,\,\,\,{\mbox{Does not exist}}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\mathop {\lim }\limits_{n \to \infty } \frac{4}{5} = \frac{4}{5}\]Shuning uchun, bu ikki qatorning har biri Divergensiya testi bilan ajralib turadi.
6-qadam
Konvergentsiya oralig'i quyida keltirilgan va xulosa qilish uchun yaqinlashuv radiusi ham ko'rsatilgan.
\[\require{bbox} \bbox[2pt,border:1px solid black]{{{\rm{Interval : }} - \frac{{53}}{{16}} < x < - \frac{{43}}{{16}}}}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\require{bbox} \bbox[2pt,border:1px solid black]{{R = \frac{5}{{16}}}}\]