AP Calculus BC / Series / Question No. 8

8. Which of the following are true? I. If \( \sum_{n=1}^{\infty} a_n \) diverges and \( 0 \le a_n \le b_n \) for all \(n\), then \( \sum_{n=1}^{\infty} b_n \) diverges. II. The sum of the geometric series \[ \frac{4}{3}+\frac{4}{9}+\frac{4}{27}+\frac{4}{81}+\cdots \] is \(2\). III. If \( \sum a_k \) diverges then \[ \lim_{k\to\infty} a_k \ne 0 \]

I only

II only

III only

I and II only

Answer is: option4

I and II only

Solution:

I. Direct Comparison Test True II. \[ a=\frac{4}{3},\quad r=\frac{1}{3},\quad \frac{a}{1-r}=\frac{\frac{4}{3}}{1-\frac{1}{3}}=2 \] True III. Harmonic series \[ \sum_{n=1}^{\infty}\frac{1}{n} \] diverges and \[ \lim_{n\to\infty}\frac{1}{n}=0 \] False

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