AP Calculus BC : 10.8 Ratio Test for  Convergence- Exam Style questions with Answer- MCQ

Question

 Which of the following series converge?

I.\(\sum_{n=1}^{\infty }\frac{n}{n+2}  \)                       II.\(\sum_{n=1}^{\infty }\frac{cos(n\pi)}{n} \)                                                       III.\(\sum_{n=1}^{\infty }\frac{1}{n}\)

(A) None
(B) II only
(C) III only
(D) I and II only
(E) I and III only

Answer/Explanation

Ans:B

I. Divergent. The limit of the nth term is not zero.

II. Convergent. This is the same as the alternating harmonic series.

 

Question

If the nth partial sum of a series \(\sum_{i=1}^{\infty }a_{i}\) is \(s_{n}=\frac{3n+1}{2n-5}\) find \(a_{5}\).

(A) \(\frac{3}{2}\)
(B) \(\frac{17}{5}\)
(C) \(-\frac{17}{15}\)
(D) \(\frac{16}{5}\)

Answer/Explanation

Ans:(C)

 

Question

Express \(1.\bar{312}\) as a ratio of two integers.
(A) \(\frac{417}{495}\)
(B) \(\frac{1049}{990}\)
(C) \(\frac{559}{495}\)
(D) \(\frac{433}{330}\)

Answer/Explanation

Ans:(D)

Question

What will the series \(\sum_{n=0}^{\infty }\frac{4^{n}}{5^{n}+1}\) do?
(A) It will converge because the ratio of consecutive terms is \(\frac{1}{2}\)
(B) It will converge because the ratio of consecutive terms is \(\frac{4}{5}\)
(C) It will converge conditionally because the ratio of consecutive terms is \(\frac{\infty }{\infty }=1\)
(D) It will diverge.

Answer/Explanation

Ans:(B)

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