Comprehensive Guide to Sequences, Series, and Taylor Expansions

Definition of a Sequence: A sequence is denoted as $\{f(n)\}_{n=k}^{\infty}$ and represents an ordered list of terms: $f(k), f(k+1), \dots, f(n), \dots$
Basic Limit Rules for Sequences:
- $\lim_{n \rightarrow \infty} \sqrt[n]{n} = 1$
- $\lim_{n \rightarrow \infty} \frac{\ln(n)}{n} = 0$
- $\lim_{n \rightarrow 0} \frac{\sin(n)}{n} = 1$
- $\lim_{n \rightarrow \infty} x^n = 0$ , provided that $|x| < 1$
- $\lim_{n \rightarrow \infty} (1 + \frac{x}{n})^n = e^x$
- $\lim_{n \rightarrow \infty} x^{\frac{1}{n}} = 1$ , provided that $x > 0$
- $\lim_{n \rightarrow \infty} \frac{x^n}{n!} = 0$

Geometric Series:
- General Form: $S_n = \sum_{n=0}^{\infty} a(r)^n$
- Conditions for Convergence: A geometric series converges if and only if $|r| < 1$ .
- Sum to Infinity: $S_{\infty} = \frac{a}{1 - r}$
- Variables:
  - $a$ : Representing the first term of the series.
  - $r$ : Representing the multiplicative factor (common ratio).
Telescoping Series:
- Evaluation Method: Plug in numbers sequentially and evaluate the partial sums to see which terms cancel out.

$n$ -th Term Test (Divergence Test):
- Usage: This should always be the first test attempted.
- Condition: Evaluate $\lim_{n \rightarrow \infty} a_n$ .
- Conclusion: If $\lim_{n \rightarrow \infty} a_n \neq 0$ , then the series $\sum a_n$ diverges.
- Critical Limitation: This test can only be used to show a series diverges. It cannot be used to prove convergence.
Integral Test:
- Conditions: The function $f(x)$ must be continuous, positive, and decreasing.
- Conclusion: The series $\sum_{n=k}^{\infty} f(n)$ and the improper integral $\int_{k}^{\infty} f(x) \, dx$ either both converge or both diverge.
$p$ -Series Test:
- Form: $\sum_{n=1}^{\infty} \frac{1}{n^p}$
- Conclusion: The series converges if and only if $p > 1$ .
Comparison Test:
- Condition: Assume $0 \leq a_n \leq b_n$ .
- Convergence Conclusion: If the larger series $\sum_{n=k}^{\infty} b_n$ converges, then the smaller series $\sum_{n=k}^{\infty} a_n$ also converges.
- Divergence Conclusion: If the smaller series $\sum_{n=k}^{\infty} a_n$ diverges, then the larger series $\sum_{n=k}^{\infty} b_n$ also diverges.
- Strategy: Find the essential behavior of the sequence and use it to bound the sequence from above (to show convergence) or below (to show divergence).

Limit Comparison Test:
- Conditions: Both $a_n > 0$ and $b_n > 0$ . Define $L = \lim_{n \rightarrow \infty} \frac{a_n}{b_n}$ .
- Conclusion 1: If $L > 0$ , then both series $S_a$ and $S_b$ either both converge or both diverge.
- Conclusion 2: If $L = 0$ and the comparison series $S_b$ converges, then the given series $S_a$ converges.
- Conclusion 3: If $L = \infty$ and the comparison series $S_b$ diverges, then the given series $S_a$ diverges.
- Definition: $a_n$ is the given function; $b_n$ is the comparison function.
Ratio Test:
- Conditions: $a_n > 0$ . Define $\rho = \lim_{n \rightarrow \infty} |\frac{a_{n+1}}{a_n}|$ .
- Conclusion:
  - If $\rho < 1$ , the series $S_n$ converges.
  - If $\rho > 1$ , the series $S_n$ diverges.
  - If $\rho = 1$ , the test is inconclusive.
- Best Use Case: Particularly effective for series involving factorials.
Root Test:
- Conditions: $a_n > 0$ . Define $\rho = \lim_{n \rightarrow \infty} (a_n)^{\frac{1}{n}}$ .
- Conclusion:
  - If $\rho < 1$ , the series $S_n$ converges.
  - If $\rho > 1$ , the series $S_n$ diverges.
  - If $\rho = 1$ , the test is inconclusive.
- Best Use Case: Effective when the general term $a_n$ is raised to the power of $n$ .

Alternating Series Test:
- Applied to series of the form $\sum_{n=N}^{\infty} (-1)^n a_n$ .
- Criteria for Convergence:
  1. $a_n$ is positive.
  2. $a_{n+1} \leq a_n$ (the terms are non-increasing).
  3. $\lim_{n \rightarrow \infty} a_n = 0$ (terms decrease to zero).
- Important Warning: Be cautious of expressions like $\sin(n\pi)$ or $\cos(n\pi)$ , as they may also cause a series to alternate.
Types of Convergence:
- Absolute Convergence: Occurs if the series of absolute values $|S_n|$ converges.
- Conditional Convergence: Occurs if the original series $S_n$ converges, but the series of absolute values $|S_n|$ diverges.
- Absolute Divergence: Occurs if the original series $S_n$ diverges.

Power Series Calculus:
- Differentiation: $\frac{d}{dx} (\sum_{n=0}^{\infty} c_n (x - a)^n) = \sum_{n=0}^{\infty} c_n n (x - a)^{n-1}$
- Integration: $\int (\sum_{n=0}^{\infty} c_n (x - a)^n) \, dx = \sum_{n=0}^{\infty} \frac{c_n}{n + 1} (x - a)^{n + 1} + C$
- Constant of Integration: To find $C$ , evaluate the expression at $x = a$ .
Error Analysis (Integral Test Estimation):
- The error, defined as the difference between the sum to infinity and the partial sum of $N$ terms, is bounded as follows:
- $\int_{N+1}^{\infty} f(x) \, dx \leq |\sum_{n=k}^{\infty} f(n) - \sum_{n=k}^{N} f(n)| \leq \int_{N}^{\infty} f(x) \, dx$

Taylor Series Definition: The Taylor series for a function $f(x)$ centered at $x = a$ is:
- $P_k(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \dots + \frac{f^{(k)}(a)}{k!}(x - a)^k$
- Maclaurin Series: A Taylor series where the center $a = 0$ .
Taylor Polynomial Orders:
- Order 0: $f(a)$
- Order 1: $f'(a)(x-a)$
- Order 2: $\frac{f''(a)}{2}(x-a)^2$
- Order 3: $\frac{f'''(a)}{3!}(x-a)^3$
Error Approximation (Taylor Remainder):
- $|\text{ACTUAL} - \text{ESTIMATE}| = |\text{ERROR}|$
- $|f(x) - P_n(x)| = |R_n(x)| \leq \frac{M |x - a|^{n+1}}{(n + 1)!}$
- Determination of $M$ : $M = \max |f^{(n+1)}(t)|$ for all $t$ between $x$ and $a$ .

Geometric Representative: $\frac{1}{1 - x} = \sum_{k=0}^{\infty} x^k$ , valid for $|x| < 1$
Sine Function: $\sin(x) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k + 1)!} x^{2k+1}$
Exponential Function: $e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}$
Cosine Function: $\cos(x) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k)!} x^{2k}$
Binomial Series: $(1 + x)^{\alpha} = \sum_{k=0}^{\infty} \binom{\alpha}{k} x^k$ , valid for $|x| < 1$
- The binomial coefficient is defined as: $\binom{\alpha}{k} = \frac{\alpha_k}{k!} = \frac{\alpha(\alpha - 1)(\alpha - 2) \dots (\alpha - (k - 1))}{k!}$
Specific Square Root Form: $\frac{1}{\sqrt{1 + x}} = \sum_{k=0}^{\infty} \frac{(-1)^k (2k)!}{4^k (k!)^2} x^k$
Inverse Tangent: $\tan^{-1}(x) = \sum_{k=0}^{\infty} \frac{(-1)^k x^{2k+1}}{2k + 1}$ , valid for $|x| \leq 1$
Natural Logarithm: $\ln(1 + x) = \sum_{k=1}^{\infty} \frac{(-1)^{k-1} x^k}{k}$ , valid for $-1 < x \leq 1$