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
