الاختبار الاول هو الـ Ratio Test : يستخدم فى تحديد الـ absolute convergence
الاختبار الثانى هو الـ Root Test: معيار الـ convergence لسلسلة لانهائية ويعتمد على الكمية
The Ratio Test
(i) If \(\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=L<1,\) then the series \(\sum_{n=1}^{\infty} a_{n}\) is absolutely convergent
(and therefore convergent)
(ii) If \(\lim|\frac{ a_{n+1}}{a_{n}} |=L>1\) or \(\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\infty,\) then the series \(\sum_{n=1}^{\infty} a_{n}\) is divergent.
(iii) If \(\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=1,\) the Ratio Test is inconclusive; that is, no conclusion
can be drawn about the convergence or divergence of \(\Sigma a_{n}\)
The Root Test
(i) If \(\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}=L<1,\) then the series \(\sum_{n=1}^{\infty} a_{n}\) is absolutely convergent
(and therefore convergent).
(ii) If \(\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}=L>1\) or \(\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}=\infty,\) then the series \(\sum_{n=1}^{\infty} a_{n}\) is
divergent.
(iii) If \(\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}=1,\) the Root Test is inconclusive.
نوعين من الاختبارات المهمة جدا فى الـ calculus
الاختبار الاول هو الـ Ratio Test : يستخدم فى تحديد الـ absolute convergence
الاختبار الثانى هو الـ Root Test : معيار الـ convergence لسلسلة لانهائية ويعتمد على الكمية
The Ratio Test
(i) If \(\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=L<1,\) then the series \(\sum_{n=1}^{\infty} a_{n}\) is absolutely convergent
(and therefore convergent)
(ii) If \(\lim|\frac{ a_{n+1}}{a_{n}} |=L>1\) or \(\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\infty,\) then the series \(\sum_{n=1}^{\infty} a_{n}\) is divergent.
(iii) If \(\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=1,\) the Ratio Test is inconclusive; that is, no conclusion
can be drawn about the convergence or divergence of \(\Sigma a_{n}\)
The Root Test
(i) If \(\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}=L<1,\) then the series \(\sum_{n=1}^{\infty} a_{n}\) is absolutely convergent
(and therefore convergent).
(ii) If \(\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}=L>1\) or \(\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}=\infty,\) then the series \(\sum_{n=1}^{\infty} a_{n}\) is
divergent.
(iii) If \(\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}=1,\) the Root Test is inconclusive.
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