Calculus BAbsolute Convergence and The Ratio and Root Tests

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.