AP Calculus BC : 10.1 Defining Convergent and Divergent Infinite  Series- Exam Style questions with Answer- MCQ

Question

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

(A) −1≤ x< 1                           (B) −1 ≤ x≤ 1                             (C) 0< 2 < x                                  (D) 0 ≤  x <2                    (E) 0 ≤ x ≤ 2x

Answer/Explanation

Ans:D

The center is x = 1, so only C, D, or E are possible. Check the endpoints.

At\(x=o:\sum_{n=1}^{\infty }\frac{(-1)^n}{n} \) converges by alternating series test.

 

Question

. Which of the following series diverge?

I.\(\sum_{k=3}^{\infty }\frac{2}{k^{2}+1}\)

II.\(\sum_{k=1}^{\infty }\left ( \frac{6}{7} \right )^{k}\)

III.\(\sum_{k=2}^{\infty }\frac{(-1)^{k}}{k}\)

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

Answer/Explanation

Ans:A

I. Compare with p-series, p=2

II. Geometric series with \(r=-\frac{6}{7}\)

 

Question

If \(\lim_{b\rightarrow \infty }\int_{1}^{b}\frac{dx}{x^p}\) is finite, then which of the following must be true?

(A)\(\sum_{n=1}^{\infty}\frac{1}{n^p}\) converges

(B)\(\sum_{n=1}^{\infty }\frac{1}{n^p}\) diverges

(C) \(\sum_{N=1}^{\infty }\frac{1}{n^{p-2}}\) converges

(D)\(\sum_{n=1}^{\infty }\frac{1}{n^{p-1}}\) converges

(E) \(\sum_{n=1}^{\infty }\frac{1}{n^{p+1}}\)diverges

Answer/Explanation

Ans:A

 

Question

Which of the following statements concerning the sequence \(\left \{ a_{n} \right \}=\frac{n}{2n^{2}-3}\) is true?

(A) Both \(\left \{ a_{n} \right \}\) and and \(\sum _{n=1}^{\infty }a_{n}\) are convergent.
(B) \(\left \{ a_{n} \right \}\) is convergent, but \(\sum_{n=1}^{\infty }a_{n}\) is divergent.
(C)\(\left \{ a_{n} \right \}\)  is divergent, but \(\sum_{n=1}^{\infty }a_{n}\) is convergent.
(D) Both \(\left \{ a_{n} \right \}\) and an \(\sum_{n=1}^{\infty }a_{n}\) are divergent.

Answer/Explanation

Ans:(B)

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