where \(R_n\) and \(L_n\) are the right- and left-hand Riemann sums for the regular \(n\)-partition.

Theorem 2 (Properties of Integrals). Assume \(f\) and \(g\) are integrable on \([a,b]\). Then:

(i) \(\int_a^b c\,f(t)\,dt = c\int_a^b f(t)\,dt\) for any \(c \in \mathbb{R}\).

(ii) \(\int_a^b (f+g)(t)\,dt = \int_a^b f(t)\,dt + \int_a^b g(t)\,dt\).

(iii) If \(m \le f(t) \le M\) for all \(t \in [a,b]\), then \(m(b-a) \le \int_a^b f(t)\,dt \le M(b-a)\).

(iv) If \(0 \le f(t)\) on \([a,b]\), then \(0 \le \int_a^b f(t)\,dt\).

(v) If \(g(t) \le f(t)\) on \([a,b]\), then \(\int_a^b g(t)\,dt \le \int_a^b f(t)\,dt\).

(vi) \(|f|\) is integrable on \([a,b]\) and \(\left|\int_a^b f(t)\,dt\right| \le \int_a^b |f(t)|\,dt\).

Theorem 3 (\(p\)-Test for Type I Improper Integrals). The integral \(\int_1^\infty \frac{1}{x^p}\,dx\) converges if and only if \(p > 1\). When \(p > 1\), \(\int_1^\infty \frac{1}{x^p}\,dx = \frac{1}{p-1}\).

Theorem 4 (Properties of Type I Improper Integrals). If \(\int_a^\infty f(x)\,dx\) and \(\int_a^\infty g(x)\,dx\) both converge, then:

1. \(\int_a^\infty cf(x)\,dx = c\int_a^\infty f(x)\,dx\).

2. \(\int_a^\infty (f+g)(x)\,dx = \int_a^\infty f(x)\,dx + \int_a^\infty g(x)\,dx\).

3. If \(f(x) \le g(x)\) for \(x \ge a\), then \(\int_a^\infty f(x)\,dx \le \int_a^\infty g(x)\,dx\).

4. If \(a < c < \infty\), then \(\int_c^\infty f(x)\,dx\) converges and \(\int_a^\infty f(x)\,dx = \int_a^c f(x)\,dx + \int_c^\infty f(x)\,dx\).

Theorem 5 (Monotone Convergence Theorem for Functions). If \(f\) is non-decreasing on \([a,\infty)\):

1. If \(\{f(x) \mid x \in [a,\infty)\}\) is bounded above, then \(\lim_{x\to\infty}f(x) = \text{lub}(\{f(x)\})\).

2. If not bounded above, then \(\lim_{x\to\infty}f(x) = \infty\).

Theorem 6 (Comparison Test for Type I Improper Integrals). Suppose \(0 \le g(x) \le f(x)\) for all \(x \ge a\) and both are continuous on \([a,\infty)\).

1. If \(\int_a^\infty f(x)\,dx\) converges, then \(\int_a^\infty g(x)\,dx\) converges.

2. If \(\int_a^\infty g(x)\,dx\) diverges, then \(\int_a^\infty f(x)\,dx\) diverges.

Theorem 7 (Absolute Convergence Theorem for Improper Integrals). If \(\int_a^\infty |f(x)|\,dx\) converges, then \(\int_a^\infty f(x)\,dx\) converges. In particular, if \(0 \le |f(x)| \le g(x)\) and \(\int_a^\infty g(x)\,dx\) converges, then \(\int_a^\infty f(x)\,dx\) converges.

Theorem 8 (\(p\)-Test for Type II Improper Integrals). The integral \(\int_0^1 \frac{1}{x^p}\,dx\) converges if and only if \(p < 1\). When \(p < 1\), \(\int_0^1 \frac{1}{x^p}\,dx = \frac{1}{1-p}\).

where the integrating factor is \(I(x) = e^{-\int f(x)\,dx}\).

Theorem 2 (Existence and Uniqueness). If \(f(x,y)\) and \(\frac{\partial f}{\partial y}\) are continuous on a rectangle containing \((x_0, y_0)\), then the IVP \(y' = f(x,y)\), \(y(x_0) = y_0\) has a unique solution on some interval containing \(x_0\).

Theorem 1 (Geometric Series Test). The geometric series \(\sum_{n=0}^\infty r^n\) converges if and only if \(|r| < 1\). When it converges, \(\sum_{n=0}^\infty r^n = \frac{1}{1-r}\).

Theorem 2 (Divergence Test). If \(\sum_{n=1}^\infty a_n\) converges, then \(\lim_{n\to\infty} a_n = 0\). Equivalently, if \(\lim_{n\to\infty} a_n \ne 0\), the series diverges.

Theorem 3 (Arithmetic for Series I). If \(\sum a_n\) and \(\sum b_n\) both converge, then:

1. \(\sum ca_n = c\sum a_n\) for every \(c \in \mathbb{R}\).

2. \(\sum(a_n + b_n) = \sum a_n + \sum b_n\).

Theorem 4 (Arithmetic for Series II).

1. If \(\sum_{n=1}^\infty a_n\) converges, then \(\sum_{n=j}^\infty a_n\) converges for each \(j\).

2. If \(\sum_{n=j}^\infty a_n\) converges for some \(j\), then \(\sum_{n=1}^\infty a_n\) converges.

In either case, \(\sum_{n=1}^\infty a_n = a_1 + a_2 + \cdots + a_{j-1} + \sum_{n=j}^\infty a_n\).

Theorem 5 (Monotone Convergence Theorem). Let \(\{a_n\}\) be non-decreasing.

1. If bounded above, then \(\{a_n\}\) converges to \(L = \text{lub}(\{a_n\})\).

2. If not bounded above, then \(\{a_n\}\) diverges to \(\infty\).

Theorem 6 (Comparison Test for Series). Assume \(0 \le a_n \le b_n\) for each \(n\).

1. If \(\sum b_n\) converges, then \(\sum a_n\) converges.

2. If \(\sum a_n\) diverges, then \(\sum b_n\) diverges.

Theorem 7 (Limit Comparison Test). Let \(a_n > 0\) and \(b_n > 0\). Suppose \(\lim_{n\to\infty}\frac{a_n}{b_n} = L\).

1. If \(0 < L < \infty\), then \(\sum a_n\) converges if and only if \(\sum b_n\) converges.

2. If \(L = 0\) and \(\sum b_n\) converges, then \(\sum a_n\) converges.

3. If \(L = \infty\) and \(\sum a_n\) converges, then \(\sum b_n\) converges.

Theorem 8 (Integral Test). Let \(f\) be continuous, positive, and decreasing on \([1,\infty)\), with \(a_n = f(n)\). Then:

(i) \(\int_1^{n+1} f(x)\,dx \le S_n \le a_1 + \int_1^n f(x)\,dx\).

(ii) \(\sum_{k=1}^\infty a_k\) converges if and only if \(\int_1^\infty f(x)\,dx\) converges.

(iii) If the series converges to \(S\), then \(\int_{n+1}^\infty f(x)\,dx \le S - S_n \le \int_n^\infty f(x)\,dx\).

Theorem 9 (\(p\)-Series Test). The series \(\sum_{n=1}^\infty \frac{1}{n^p}\) converges if and only if \(p > 1\).

Theorem 10 (Alternating Series Test). If (1) \(a_n > 0\), (2) \(a_{n+1} \le a_n\), and (3) \(\lim_{n\to\infty} a_n = 0\), then \(\sum_{n=1}^\infty (-1)^{n-1}a_n\) converges. Moreover, \(|S_k - S| \le a_{k+1}\).

Theorem 11 (Absolute Convergence Theorem). If \(\sum |a_n|\) converges, then \(\sum a_n\) converges.

Theorem 12 (Rearrangement Theorem).

1. If \(\sum a_n\) converges absolutely and \(\sum b_n\) is any rearrangement, then \(\sum b_n\) converges and \(\sum b_n = \sum a_n\).

2. If \(\sum a_n\) converges conditionally, then for any \(\alpha \in \mathbb{R}\) (or \(\alpha = \pm\infty\)), there exists a rearrangement \(\sum b_n\) with \(\sum b_n = \alpha\).

Theorem 13 (Ratio Test). Given \(\sum a_n\), suppose \(\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| = L\).

1. If \(0 \le L < 1\), the series converges absolutely.

2. If \(L > 1\), the series diverges.

3. If \(L = 1\), no conclusion is possible.

Theorem 14 (Polynomial vs. Factorial Growth). For any \(x \in \mathbb{R}\), \(\lim_{n\to\infty}\frac{x^n}{n!} = 0\).

Theorem 1 (Fundamental Convergence Theorem for Power Series). Given \(\sum a_n(x-a)^n\) with radius \(R\):

1. If \(R = 0\), the series converges only at \(x = a\).

2. If \(0 < R < \infty\), the series converges absolutely for \(|x-a| < R\) and diverges for \(|x-a| > R\).

3. If \(R = \infty\), the series converges absolutely for all \(x \in \mathbb{R}\).

The interval of convergence may or may not include the endpoints.

Theorem 2 (Test for Radius of Convergence). If \(\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| = L\), then:

1. If \(0 < L < \infty\), then \(R = \frac{1}{L}\).

2. If \(L = 0\), then \(R = \infty\).

3. If \(L = \infty\), then \(R = 0\).

Theorem 3 (Equivalence of Radius of Convergence). If \(p\) and \(q\) are non-zero polynomials with \(q(n) \ne 0\) for \(n \ge k\), then \(\sum a_n(x-a)^n\) and \(\sum a_n\frac{p(n)}{q(n)}(x-a)^n\) have the same radius of convergence (though possibly different intervals).

Theorem 4 (Abel’s Theorem: Continuity of Power Series). If \(f(x) = \sum a_n(x-a)^n\) on the interval of convergence \(I\), then \(f\) is continuous on \(I\).

Theorem 7 (Taylor Polynomial Properties). The Taylor polynomial \(T_{n,a}(x)\) is the unique polynomial of degree \(\le n\) such that \(T_{n,a}^{(k)}(a) = f^{(k)}(a)\) for \(k = 0, 1, \ldots, n\).

for all \(x \in [-1,1]\).

for all \(x \in I\).

Theorem 1 (Limit Theorem). \(\lim_{t\to t_0}\vec{F}(t) = (L_1, L_2)\) if and only if \(\lim_{t\to t_0}x(t) = L_1\) and \(\lim_{t\to t_0}y(t) = L_2\).

Theorem 2 (Continuity Theorem). \(\vec{F}(t) = (x(t), y(t))\) is continuous at \(t_0\) if and only if both \(x(t)\) and \(y(t)\) are continuous at \(t_0\).

Theorem 3 (Differentiation Theorem). \(\vec{F}(t) = (x(t), y(t))\) is differentiable at \(t_0\) if and only if both \(x(t)\) and \(y(t)\) are differentiable at \(t_0\), and \(\vec{F}'(t_0) = (x'(t_0), y'(t_0))\).