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)