Show all supporting work on scratch paper. State clearly which convergence test you use and verify its conditions before applying it.请在草稿纸上写出所有解题步骤。使用某判别法前，须明确说明所用判别法并验证其条件。
Which of the following is a true statement about the harmonic series $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n}$?下列关于调和级数 $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n}$ 的说法中，哪个正确？
(A)it converges to $\ln 2$它收敛到 $\ln 2$
(B)it converges by the integral test由积分判别法收敛
(C)it diverges, even though its terms $\to 0$它发散，尽管各项 $\to 0$
Q8EASY10.6 Direct Comparison Test10.6 比较判别法No Calculator
Since $0\le\dfrac{1}{n^{3}+1}\le\dfrac{1}{n^{3}}$ for all $n\ge 1$, and $\displaystyle\sum\frac{1}{n^{3}}$ converges, what does the Direct Comparison Test conclude about $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{3}+1}$?因对所有 $n\ge 1$ 有 $0\le\dfrac{1}{n^{3}+1}\le\dfrac{1}{n^{3}}$，且 $\displaystyle\sum\frac{1}{n^{3}}$ 收敛，比较判别法对 $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{3}+1}$ 得出什么结论？
Q13HARD10.10 Alternating Series Error Bound10.10 交错级数误差界Calculator
For $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n!}$, what is the smallest $n$ such that the partial sum $S_n$ approximates the sum with error less than $0.001$?对 $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n!}$，使部分和 $S_n$ 的误差小于 $0.001$ 的最小 $n$ 是多少？
Let $P_3(x)$ be the 3rd-degree Taylor polynomial for $f(x)=e^{x}$ centered at $a=0$. If $|f^{(4)}(c)|\le 3$ for $0\le c\le 1$, the Lagrange error bound for $|f(1)-P_3(1)|$ is设 $P_3(x)$ 为 $f(x)=e^{x}$ 在 $a=0$ 处的 3 次泰勒多项式。若对 $0\le c\le 1$ 有 $|f^{(4)}(c)|\le 3$，则 $|f(1)-P_3(1)|$ 的拉格朗日误差界为
Q16MEDIUM10.13 Radius of Convergence10.13 收敛半径No Calculator
The radius of convergence of $\displaystyle\sum_{n=1}^{\infty}\frac{(x+1)^{n}}{n\cdot 3^{n}}$ is$\displaystyle\sum_{n=1}^{\infty}\frac{(x+1)^{n}}{n\cdot 3^{n}}$ 的收敛半径为
Free-response answers must state the test used and verify its conditions explicitly. A calculator is permitted unless marked otherwise.自由回答题须明确说明所用判别法并验证其条件。除非特别注明，否则允许使用计算器。
For each series below, state whether it converges or diverges, name the test used, and (if it converges by a direct computation) give the sum.对下列各级数，判断其是收敛还是发散，写出所用判别法，并（若可直接求和）给出和。
Let $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{2n+1}$.设 $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{2n+1}$。
(a)Show that the series converges, verifying all three conditions of the Alternating Series Test.验证交错级数判别法的三个条件，证明该级数收敛。
(b)Determine whether the series is absolutely or conditionally convergent. Justify using an appropriate test on $\displaystyle\sum\left|a_n\right|$.判断该级数是绝对收敛还是条件收敛，并对 $\displaystyle\sum\left|a_n\right|$ 用适当的判别法加以论证。
(c)Use the alternating series error bound to find the smallest $n$ such that $S_n$ approximates the sum with error less than $0.05$.利用交错级数误差界，求使 $S_n$ 的误差小于 $0.05$ 的最小 $n$。
Let $f(x)=\ln x$, and let $P_3(x)$ be the 3rd-degree Taylor polynomial for $f$ centered at $a=1$.设 $f(x)=\ln x$，$P_3(x)$ 为 $f$ 在 $a=1$ 处的 3 次泰勒多项式。
(a)Find $P_3(x)$. Show the derivatives used.求 $P_3(x)$，写出所用的各阶导数。
(b)Use $P_3(x)$ to approximate $\ln(1.2)$.用 $P_3(x)$ 近似 $\ln(1.2)$。
(c)Given that $|f^{(4)}(c)|\le 6$ for $1\le c\le 1.2$, find the Lagrange error bound for the approximation in part (b), and confirm it is consistent with the actual error ($\ln(1.2)\approx 0.18232$).已知对 $1\le c\le 1.2$ 有 $|f^{(4)}(c)|\le 6$，求 (b) 中近似值的拉格朗日误差界，并验证该界与实际误差一致（$\ln(1.2)\approx 0.18232$）。
(a)Starting from the geometric series for $\dfrac{1}{1+x}$, use term-by-term integration to find the Maclaurin series for $g(x)$.从 $\dfrac{1}{1+x}$ 的等比级数出发，用逐项积分求 $g(x)$ 的麦克劳林级数。
(b)Use the first four nonzero terms of this series to approximate $\ln(1.1)$.用该级数的前四个非零项近似 $\ln(1.1)$。
(c)Use the alternating series error bound to bound the error in the approximation in part (b).用交错级数误差界估计 (b) 中近似值的误差。
(d)State the radius of convergence of the series found in part (a), and determine whether $x=1$ is included in the interval of convergence.写出 (a) 中所求级数的收敛半径，并判断 $x=1$ 是否属于收敛区间。