Sections 1 to 7: radius and interval of convergence, power series representations, differentiation and integration of power series, Taylor and Maclaurin series, key Maclaurin library, Taylor remainder, applications第 1 至 7 节：收敛半径与收敛区间，幂级数表示，幂级数的微分与积分，泰勒级数与麦克劳林级数，常用麦克劳林公式库，泰勒余项，应用CALC II
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PART I · CORE TECHNIQUES核心技巧Computational fluency · 28 marks计算熟练度 · 28分
Radius and Interval of Convergence; Power Series Algebra收敛半径与收敛区间；幂级数运算
Apply the ratio test to find the radius of convergence, then check each endpoint separately using a named convergence test. State the interval of convergence in interval notation.用比值审敛法求收敛半径，然后用具名收敛判别法分别检验每个端点，以区间符号写出收敛区间。
Q1MEDIUMCOREratio test, radius and interval of convergence比值审敛法，收敛半径与收敛区间[8 marks]
Find the radius of convergence and the interval of convergence for each power series. Check both endpoints.求每个幂级数的收敛半径和收敛区间，并检验两个端点。
Find the radius of convergence and the interval of convergence for each power series. At each endpoint, name the test you apply and state its conclusion.求每个幂级数的收敛半径和收敛区间。在每个端点处，写出所用判别法的名称并陈述结论。
Q3MEDIUMCOREbuilding new series by substitution and integration通过换元与积分构造新级数[10 marks]
Start from the standard Maclaurin series $\dfrac{1}{1-x}=\sum_{n=0}^{\infty}x^{n}$ for $|x|<1$.从标准麦克劳林级数 $\dfrac{1}{1-x}=\sum_{n=0}^{\infty}x^{n}$（$|x|<1$）出发。
(a)By substituting $x\mapsto -x^{2}$, write $\dfrac{1}{1+x^{2}}$ as a power series and state its interval of convergence.令 $x\mapsto -x^{2}$，将 $\dfrac{1}{1+x^{2}}$ 写成幂级数并写出其收敛区间。[3]
(b)Integrate the result of (a) term by term to obtain the Maclaurin series for $\arctan x$. Include the constant of integration and state the interval of convergence.将 (a) 的结果逐项积分，得到 $\arctan x$ 的麦克劳林级数，写出积分常数并给出收敛区间。[4]
(c)Use the series in (b) evaluated at $x=1$ to write an infinite series expression for $\pi/4$. Identify the convergence test that guarantees the series converges at $x=1$.将 (b) 中的级数代入 $x=1$，写出 $\pi/4$ 的无穷级数表达式，并指出保证该级数在 $x=1$ 处收敛的判别法。[3]
PART II · DEFINITIONS AND PROOF定义与证明Rigorous arguments · 26 marks严格论证 · 26分
Taylor and Maclaurin Series from the Definition; Remainder Bounds从定义出发的泰勒级数与麦克劳林级数；余项估计
In derivation questions, compute each coefficient directly from $c_{n}=f^{(n)}(a)/n!$; do not quote a known series without justification. In remainder questions, state the Lagrange form explicitly and bound $|f^{(n+1)}(z)|$ over the relevant interval before concluding.在推导题中，直接由 $c_{n}=f^{(n)}(a)/n!$ 计算各系数，不可在未加说明的情况下引用已知级数。在余项题中，先明确写出拉格朗日余项形式，并在相关区间上对 $|f^{(n+1)}(z)|$ 给出上界，再得出结论。
Q4HARDPROOFMaclaurin series from the definition; interval of convergence proof从定义出发的麦克劳林级数；收敛区间证明[10 marks]
Let $f(x)=\ln(1+x)$.设 $f(x)=\ln(1+x)$。
(a)Compute $f(0),\,f'(0),\,f''(0),\,f'''(0),\,f^{(4)}(0)$ and hence write down the general Maclaurin series for $\ln(1+x)$ in sigma notation.计算 $f(0),\,f'(0),\,f''(0),\,f'''(0),\,f^{(4)}(0)$，并由此写出 $\ln(1+x)$ 的一般麦克劳林级数的求和符号形式。[5]
(b)Apply the ratio test to show the radius of convergence is $R=1$.用比值审敛法证明收敛半径为 $R=1$。[2]
(c)Test the endpoint $x=1$. Name the convergence test you apply and state its conclusion.检验端点 $x=1$，写出所用判别法的名称并陈述结论。[2]
(d)Test the endpoint $x=-1$. State the conclusion and write the full interval of convergence.检验端点 $x=-1$，陈述结论并写出完整的收敛区间。[1]
Q5HARDPROOFTaylor remainder (Lagrange form) and error bound泰勒余项（拉格朗日形式）与误差估计[8 marks]
Let $f(x)=e^{x}$. Let $T_{n}(x)$ denote the degree-$n$ Maclaurin polynomial of $e^{x}$.设 $f(x)=e^{x}$，$T_{n}(x)$ 表示 $e^{x}$ 的 $n$ 次麦克劳林多项式。
(a)Write down $T_{4}(x)$ and the Lagrange form of the remainder $R_{4}(x)$ at a general $x$.写出 $T_{4}(x)$ 和一般 $x$ 处余项 $R_{4}(x)$ 的拉格朗日形式。[3]
(b)Using the fact that $e^{z}<3$ for $0\le z\le 1$, show that the error $|e-T_{4}(1)|$ satisfies $|e-T_{4}(1)|<\dfrac{1}{40}$.利用 $0\le z\le 1$ 时 $e^{z}<3$ 这一事实，证明误差 $|e-T_{4}(1)|$ 满足 $|e-T_{4}(1)|<\dfrac{1}{40}$。[3]
(c)Compute $T_{4}(1)$ explicitly and verify that your error bound is plausible by comparing with the true value $e\approx 2.71828$.明确计算 $T_{4}(1)$，并与真实值 $e\approx 2.71828$ 比较，验证误差估计的合理性。[2]
Q6HARDPROOFbinomial series; radius and interval of convergence二项级数；收敛半径与收敛区间[8 marks]
The binomial series states $(1+x)^{k}=\displaystyle\sum_{n=0}^{\infty}\binom{k}{n}x^{n}$ for $|x|<1$, where $\dbinom{k}{n}=\dfrac{k(k-1)\cdots(k-n+1)}{n!}$.二项级数为 $(1+x)^{k}=\displaystyle\sum_{n=0}^{\infty}\binom{k}{n}x^{n}$（$|x|<1$），其中 $\dbinom{k}{n}=\dfrac{k(k-1)\cdots(k-n+1)}{n!}$。
(a)Use the binomial series with $k=-\tfrac{1}{2}$ to write $(1+x)^{-1/2}$ as a power series. Compute at least the first four non-zero terms and give the general coefficient.取 $k=-\tfrac{1}{2}$，利用二项级数将 $(1+x)^{-1/2}$ 写成幂级数，至少计算前四个非零项并给出一般系数。[4]
(b)Apply the ratio test to confirm the radius of convergence is $R=1$.用比值审敛法验证收敛半径为 $R=1$。[2]
(c)By substituting $x\mapsto -t^{2}$ in the result of (a), write $\dfrac{1}{\sqrt{1-t^{2}}}$ as a power series and state its interval of convergence.在 (a) 的结果中令 $x\mapsto -t^{2}$，将 $\dfrac{1}{\sqrt{1-t^{2}}}$ 写成幂级数并给出收敛区间。[2]
PART III · APPLICATIONS AND SYNTHESIS应用与综合Extended problems · 28 marks综合题 · 28分
Approximation, Definite Integrals via Series, and Limits近似计算、定积分的级数方法与极限
Set up each series representation before applying it. For definite integral approximations, state how many terms you use and apply the alternating series estimation theorem or the Lagrange remainder to guarantee the error bound. For limits, expand only as many terms as needed to resolve the indeterminate form.在应用之前先建立每个级数表示。对于定积分的近似计算，说明所用项数，并用交错级数估计定理或拉格朗日余项来保证误差界。对于极限，只展开足以消除不定型所需的项数。
Q7MEDIUMAPPLIEDseries manipulation: differentiation and multiplication级数运算：微分与乘法[8 marks]
Throughout this question you may use the standard Maclaurin series $e^{x}=\sum_{n=0}^{\infty}\dfrac{x^{n}}{n!}$ and $\sin x=\sum_{n=0}^{\infty}\dfrac{(-1)^{n}x^{2n+1}}{(2n+1)!}$.本题全程可使用标准麦克劳林级数 $e^{x}=\sum_{n=0}^{\infty}\dfrac{x^{n}}{n!}$ 和 $\sin x=\sum_{n=0}^{\infty}\dfrac{(-1)^{n}x^{2n+1}}{(2n+1)!}$。
(a)Differentiate the series for $\sin x$ term by term to recover the Maclaurin series for $\cos x$. Write the first four non-zero terms and the general term.逐项微分 $\sin x$ 的级数，得到 $\cos x$ 的麦克劳林级数，写出前四个非零项和一般项。[3]
(b)Write down the Maclaurin series for $e^{-x^{2}}$ by substituting $x\mapsto -x^{2}$ in the series for $e^{x}$.在 $e^{x}$ 的级数中令 $x\mapsto -x^{2}$，写出 $e^{-x^{2}}$ 的麦克劳林级数。[2]
(c)Hence find the Maclaurin series for $x\,e^{-x^{2}}$ and write a closed-form series for $\displaystyle\int_{0}^{t}x\,e^{-x^{2}}\,dx$ valid for all $t$.由此求 $x\,e^{-x^{2}}$ 的麦克劳林级数，并写出对所有 $t$ 成立的 $\displaystyle\int_{0}^{t}x\,e^{-x^{2}}\,dx$ 的封闭形式级数。[3]
Q8HARDAPPLIEDapproximating a definite integral with error bound带误差估计的定积分近似[10 marks]
Consider $I=\displaystyle\int_{0}^{1/2}\frac{\sin x}{x}\,dx$. (The integrand is defined to equal $1$ at $x=0$.)考虑 $I=\displaystyle\int_{0}^{1/2}\frac{\sin x}{x}\,dx$。（被积函数在 $x=0$ 处定义为 $1$。）
(a)Write $\dfrac{\sin x}{x}$ as a power series in $x$ using the Maclaurin series for $\sin x$. State its interval of convergence.利用 $\sin x$ 的麦克劳林级数，将 $\dfrac{\sin x}{x}$ 写成 $x$ 的幂级数，并写出其收敛区间。[3]
(b)Integrate the series term by term to obtain a series for $I$. Write out at least the first four terms.逐项积分该级数，得到 $I$ 的级数表示，至少写出前四项。[3]
(c)By the alternating series estimation theorem, determine the minimum number of terms needed to approximate $I$ to within $10^{-4}$, and state the approximation. Show your working.用交错级数估计定理，确定将 $I$ 近似至 $10^{-4}$ 精度所需的最少项数，给出近似值并展示计算过程。[4]
(b) $\displaystyle\lim_{x\to 0}\frac{\cos x - 1 + \tfrac{1}{2}x^{2}}{x^{4}}$ [3]
Q10HARDAPPLIEDnumerical approximation with guaranteed accuracy; remainder bound有精度保证的数值近似；余项估计[4 marks]
You wish to approximate $e^{-0.1}$ using the Maclaurin polynomial $T_{n}(-0.1)$ of $e^{x}$.用 $e^{x}$ 的麦克劳林多项式 $T_{n}(-0.1)$ 近似 $e^{-0.1}$。
(a)Using the Lagrange remainder with the bound $e^{z}<2$ for $-0.1\le z\le 0$, find the smallest $n$ such that $|e^{-0.1}-T_{n}(-0.1)|<5\times 10^{-6}$.利用拉格朗日余项及 $-0.1\le z\le 0$ 时 $e^{z}<2$ 的上界，求满足 $|e^{-0.1}-T_{n}(-0.1)|<5\times 10^{-6}$ 的最小 $n$。[3]
(b)State the value $T_{n}(-0.1)$ for that $n$, giving your answer as an exact fraction or to 7 significant figures.对该 $n$ 值，以精确分数或7位有效数字写出 $T_{n}(-0.1)$ 的值。[1]