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• Notes

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.