CALC
Pos_hnd

# Pos_hnd - Positive-Term Series Theorems on Convergence or...

• Notes
• nizhes
• 2

This preview shows pages 1–2. Sign up to view the full content.

Positive-Term Series Theorems on Convergence or Divergence of a Positive-Term Series 1. If n a is a positive-term series and if there exists a number M such that M a a a S n n < + + + = 2 1 for every n , then the series converges and has a sum M S . If no such M exists, the series diverges. 2. Integral Test If n a is a series, let n a n f = ) ( and let f be the function obtained by replacing n with x . If f is positive-valued, continuous, and decreasing for every real number 1 x , then the series n a (i) converges if 1 ) ( dx x f converges. (ii) diverges if 1 ) ( dx x f diverges. 3 . P -series If p is a positive real number, then the p -series + + + + + = p p p p n n 1 3 1 2 1 1 1 1 (i) converges if 1 p (ii) diverges if 1 p 4. Basic Comparison Test Let n a and n b be positive-term series. (i) If n b converges and n n b a for every positive integer n , then n a converges. (ii) If n b diverges and n n b a for every positive integer n , then n a diverges. 5. Limit Comparison Test Let n a and n b be positive-term series.

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: (i) If lim = ∞ → c b a n n n then either both series converge or both series diverge. (ii) If lim = ∞ → n n n b a and ∑ n b converges, then ∑ n a also converges. (iii) If ∞ = ∞ → n n n b a lim and ∑ n b diverges, then ∑ n a also diverges. 6. Ratio Test Let ∑ n a be a positive-term series, and suppose that L a a n n n = + ∞ → 1 lim (i) If 1 < L , the series is convergent (ii) If ∞ = + ∞ → n n n a a L 1 lim or 1 , the series is divergent. (iii) If 1 = L , apply a different test; the series may be convergent or divergent. 7. Root Test Let ∑ n a be a positive-term series, and suppose that L a n n n = ∞ → lim (i) If 1 < L , the series is convergent (ii) If ∞ = ∞ → n n n a L lim or 1 , the series is divergent. (iii) If 1 = L , apply a different test; the series may be convergent or divergent....
View Full Document

• Spring '08
• varies
• Calculus, lim, positive-term series

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern