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)
