AP Calculus BC : 10.6 Comparison Tests for  Convergence- Exam Style questions with Answer- MCQ

Question

Which of the following statements about convergence of the series \(\sum_{n=1}^{\infty }\frac{1}{ln(n+1)}\)  is true?

A \(\sum_{n=1}^{\infty }\frac{1}{ln(n+1)}\) converges by comparison with \(\sum_{n=1}^{\infty }\frac{1}{n}\)

B \(\sum_{n=1}^{\infty }\frac{1}{ln(n+1)}\) converges by comparison with \(\sum_{n=1}^{\infty }\frac{1}{n^2}\)

C \(\sum_{n=1}^{\infty }\frac{1}{ln(n+1)}\) diverges by comparison with \(\sum_{n=1}^{\infty }\frac{1}{n}\)

D \(\sum_{n=1}^{\infty }\frac{1}{ln(n+1)}\) diverges by comparison with \(\sum_{n=1}^{\infty }\frac{1}{n^2}\)

Answer/Explanation

 

Question 

 What are all values of x for which the series \(\sum_{n=1}^{\infty }\frac{x^{n}}{n}\)converges?
(A)\(-1\leq x\leq 1\)                             (B)\(-1<x\leq 1\)                (C)\(-1\leq x<1 \)                     (D)\(-1<x<1\)                                     (E)All real x

Answer/Explanation

Ans:C

 

Question 

 Which of the following series converge?

I. \(\sum_{n=1}^{\infty } (-1)^{n+1}\frac{1}{2n+1}\)
II.\( \sum_{n=1}^{\infty }\frac{1}{n}\left ( \frac{3}{2} \right )^{n}\)
III.\(\sum_{n=1}^{\infty }\frac{1}{nlnn}\)

(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

Answer/Explanation

Ans:A

Question

What are all values of x for which the series \(\sum_{n=1}^{\infty }\frac{(x-2)^n}{n.3^{n}}\) converges?

(A) −3≤x ≤ 3 
(B) −3< x< 3
(C) −1< x ≤ 5 
(D) −1≤x ≤ 5 
(E) −1≤x < 5

Answer/Explanation

Ans:E

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