BC Test

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241

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Mathematics

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Feb 20, 2024

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md

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4

Uploaded by MasterBraveryPorcupine32

### True or False Questions 1. The Harmonic Series $\sum_{n=1}^{\infty} \frac{1}{n}$ converges (T / F) 2. The Alternating Harmonic Series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ converges (T / F) ### Multiple Choice Questions 1. Which of the following Series converge? $I. \sum_{n=1}^{\infty}\frac{8^n}{n!}$ $II. \sum_{n=1}^{\infty}\frac{n!} {n^{100}}$ $III. \sum_{n=1}^{\infty}e^{-n}(n^2+4)$ a) I only b) II only c) I and III only d) I, II, and III 2. What does the series $\sum_{n=1}^{\infty}\frac{4n+2}{3n}$ converge to? a) $\frac{4}{3}$ b) 1 c) 0 d) Divergent 3. What is the value of $S_4$ in the sequence of partial sums $S_n$ of sequence $a_n = \frac{\pi^n}{2^n}, n>0$ a) $\frac{\pi^4}{16}$
b) $\frac{\pi^4+2\pi^3+4\pi^2+8\pi}{16}$ c) $\frac{\pi^4+2\pi^3+4\pi^2+8\pi}{32}$ d) $\frac{\pi^4+2\pi^3}{8}$ 4. If the $\lim_{n\rightarrow\infty} S_n = 1$ where $S_n$ is the sequence of partial sums of the sequence $a_n$. What does $\sum_{n=1}^{\infty}a_n$ equal? a) 1 b) 0 c) $\infty$ d) Can not be determined 5. Considering the Alternating Series $\sum_{n=0}^{\infty}(-1)^na_n$ which of the following are true? $I.$ If the series converges conditionally, any rearranged order of the series sums to the same value $II. $ If the series converges absolutely, any rearranged order of the series sums to the same value $III. $ If the series $\sum_{n=0}^{\infty}a_n$ converges, then $\sum_{n=0}^{\ infty}(-1)^na_n$ converges a) I only b) II only c) II and III only d) I and III only ### Free Response Questions 1. All questions in this section refer to the series $\sum_{n=0}^{\infty} \frac{(- 1)^n}{2n+1}$ a) Determine whether the given series converges or not
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