Calculus II: Sequences, Infinite Series, and Factorial Limits Study Strategies

Properties and Simplification of Factorials

• Definition of Factorials: The factorial of a number $n$, denoted by $n!$, is defined as the product of all positive integers less than or equal to $n$. For example, $4! = 4 \times 3 \times 2 \times 1$.
• Recursive Property of Factorials:
  • One can take the first factor off the front of a factorial: $n! = n \times (n-1)!$.
  • Example: $4! = 4 \times 3!$.
  • One can also take the first two factors off: $n! = n \times (n-1) \times (n-2)!$.
  • Example: $4! = 4 \times 3 \times 2!$.
  • In general, $n! = n \times (n-1) \times (n-2) \times \dots \times 3 \times 2 \times 1$.
• Simplifying Factorial Expressions: To simplify expressions involving factorials in the numerator and denominator, identify the larger factorial and expand it until it matches the smaller one.
  • Example Problem: Find the limit of $\frac{(2n-1)!}{(2n+1)!}$ as $n \rightarrow \infty$.
  • Step 1: Compare $(2n-1)$ and $(2n+1)$. Since $(2n+1)$ is larger, expand it: $(2n+1)! = (2n+1) \times (2n) \times (2n-1)!$.
  • Step 2: Substitute back into the expression: $\frac{(2n-1)!}{(2n+1) \times (2n) \times (2n-1)!}$.
  • Step 3: Cancel the common factor $(2n-1)!$, resulting in $\frac{1}{(2n+1) \times (2n)}$.
  • Step 4: Take the limit: $\lim_{n \rightarrow \infty} \frac{1}{2n(2n+1)} = 0$, because the numerator is fixed at 1 and the denominator grows infinitely large.

Sequences vs. Series

• Sequence: An ordered list of numbers.
  • Example: $a_n = \frac{1}{n}$ generates the sequence $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots$.
• Series (or Sums): The sum of the terms of a sequence.
  • Example: $\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \dots$.
• Sequence of Terms vs. Sequence of Partial Sums:
  • $a_n$ refers to the individual terms in the list.
  • $s_n$ refers to the sum of the first $n$ terms.

Limits of Alternating Sequences

• Alternating Terms: Sequences containing factors like $(-1)^n$ alternate in sign between positive and negative values. These can be difficult to evaluate using standard tools like L'Hôpital's Rule.
• Absolute Value Theorem for Limits: If $\lim_{n \rightarrow \infty} |a_n| = 0$, then $\lim_{n \rightarrow \infty} a_n = 0$.
  • Problem Application (Problem 41): Find the limit of $a_n = \frac{(-1)^n}{2\sqrt{n}}$.
  • Take the absolute value: $|a_n| = \frac{1}{2\sqrt{n}}$.
  • The limit of the absolute value is $\lim_{n \rightarrow \infty} \frac{1}{2\sqrt{n}} = 0$.
  • Therefore, the limit of the original alternating sequence is also 0.
• Squeeze Theorem Proof for Alternating Sequences:
  • State the inequality: $-\frac{1}{2\sqrt{n}} \le \frac{(-1)^n}{2\sqrt{n}} \le \frac{1}{2\sqrt{n}}$.
  • Since $\lim_{n \rightarrow \infty} (-\frac{1}{2\sqrt{n}}) = 0$ and $\lim_{n \rightarrow \infty} \frac{1}{2\sqrt{n}} = 0$, the middle term must also limit to 0.

Evaluating Indeterminate Forms in Sequences

• Indeterminate Power Form ($1^{\infty}$): For a sequence like $a_n = (1 + \frac{2}{n})^n$, both the base and the exponent change as $n \rightarrow \infty$.
  • Method of Solution: Use the identity $x = e^{\ln(x)}$. Rewrite the sequence as $e^{n \ln(1 + \frac{2}{n})}$.
  • Because the exponential function is continuous, move the limit into the exponent: $e^{\lim_{n \rightarrow \infty} n \ln(1 + \frac{2}{n})}$.
  • The exponent limit is of the form $\infty \times 0$. Rewrite as a fraction: $\frac{\ln(1 + \frac{2}{n})}{\frac{1}{n}}$, which is now a $\frac{0}{0}$ form.
  • Apply L'Hôpital's Rule (treating $n$ as a continuous variable $x$):
    • Derivative of numerator: $\frac{1}{1 + \frac{2}{n}} \times (-2n^{-2})$.
    • Derivative of denominator: $-n^{-2}$.
    • The terms $-n^{-2}$ cancel, leaving $\lim_{n \rightarrow \infty} \frac{2}{1 + \frac{2}{n}} = 2$.
  • Final Answer: $e^2$.
• Indeterminate Difference Form ($\infty - \infty$): For the sequence $a_n = \ln(2n^2+1) - \ln(n^2+1)$.
  • Apply the logarithmic power rule: $\ln(A) - \ln(B) = \ln(\frac{A}{B})$.
  • The sequence becomes $\ln(\frac{2n^2+1}{n^2+1})$.
  • Since $\ln$ is continuous on its domain, take the limit of the fraction inside: $\lim_{n \rightarrow \infty} \frac{2n^2+1}{n^2+1} = 2$.
  • Final Answer: $\ln(2)$.

Continuity and Limits

• Continuous Functions in Calculus: Most standard functions are continuous on their domains, including polynomials, rational functions, root functions, trigonometric functions, inverse trigonometric functions, logarithmic functions, and exponential functions.
• Limit of Composition: If $f$ is continuous, then $\lim_{n \rightarrow \infty} f(g(n)) = f(\lim_{n \rightarrow \infty} g(n))$.
  • Example: $\lim_{x \rightarrow \infty} \cos(\frac{1}{x}) = \cos(\lim_{x \rightarrow \infty} \frac{1}{x}) = \cos(0) = 1$.

Sequence Monotonicity and Boundedness

• Increasing Sequence: A sequence where $a_n \le a_{n+1}$ for all $n$ (each term is greater than or equal to the predecessor).
• Decreasing Sequence: A sequence where $a_n \ge a_{n+1}$ for all $n$ (each term is less than or equal to the predecessor).
• Monotonic: A sequence that is either entirely increasing or entirely decreasing.
• Boundedness:
  • Bounded Above: There exists a number $M$ such that $a_n \le M$ for all $n$.
  • Bounded Below: There exists a number $m$ such that $a_n \ge m$ for all $n$.
  • Least Upper Bound (LUB): The smallest possible upper bound for a set.
  • Greatest Lower Bound (GLB): The largest possible lower bound for a set.
• Monotonic Sequence Theorem: Every bounded, monotonic sequence is convergent (it must have a limit).
• Testing Monotonicity using Calculus: Embed the sequence in a function $f(x)$ such that $f(n) = a_n$.
  • Check the derivative $f'(x)$.
  • If $f'(x) > 0$, the function and sequence are increasing.
  • If $f'(x) < 0$, the function and sequence are decreasing.

Geometric Sequences and Series

• Geometric Sequence Form: $a_n = r^n$.
• Convergence Conditions for $r^n$:
  • If $|r| < 1$, $\lim_{n \rightarrow \infty} r^n = 0$.
  • If $|r| > 1$, the sequence is divergent (oscillates or grows infinitely).
  • If $r = 1$, the sequence is constant and converges to 1.
• Geometric Series Formula: $\sum_{n=1}^{\infty} a r^{n-1} = a + ar + ar^2 + \dots$
  • Converges if and only if $|r| < 1$.
  • Sum of the series: $S = \frac{a}{1-r}$.
  • Example (Problem 80 context): $\sum_{n=1}^{\infty} 5(\frac{2}{3})^{n-1}$.
    • $a = 5$, $r = \frac{2}{3}$.
    • Since $|\frac{2}{3}| < 1$, it converges to $\frac{5}{1 - \frac{2}{3}} = \frac{5}{\frac{1}{3}} = 15$.

Infinite Series and Partial Sums

• Definition: An infinite series is the sum of an infinite list of numbers $\sum_{n=1}^{\infty} a_n$.
• Sequence of Partial Sums ($s_n$):
  • $s_1 = a_1$
  • $s_2 = a_1 + a_2$
  • $s_3 = a_1 + a_2 + a_3$
• Recurrence Relation for Partial Sums: $s_n = s_{n-1} + a_n$. The sum of the first $n$ terms is the sum of the first $n-1$ terms plus the next term.
• Convergence of a Series: A series converges if the sequence of its partial sums $\lim_{n \rightarrow \infty} s_n$ converges to a finite limit $L$. If the limit is infinite or does not exist, the series diverges.
• Test for Divergence: If $\lim_{n \rightarrow \infty} a_n \neq 0$, then the series $\sum a_n$ must diverge. Note: If the limit is 0, the series might still diverge (e.g., the Harmonic Series).
• The Harmonic Series: $\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \dots$ diverges, even though individual terms go to 0.
• The p-series Example: $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges because the terms get small fast enough.

Calculator Procedures (TI-84)

• Sequence Mode: Change the calculator mode to "Sequence".
• Entering Sequences: Under $Y=$, enter the formula for $u(n)$.
• Graphing Partial Sums: Define $u(n) = u(n-1) + a_n$.
  • The "u" is accessed using $2nd + 7$.
  • The variable "n" is the standard variable key.
  • Set the starting value (e.g., $u(nMin) = a_1$).
• Summation Math Menu: Use the "Summation" command ($\Sigma$) under the Math menu for modern TI-84 calculators to calculate specific finite sums.

Questions & Discussion

• Student Question: "What is the limit of the sequence $\frac{(-1)^n}{2\sqrt{n}}$?"
  • Response: The professor explained using the absolute value theorem and the Squeeze Theorem to show the limit is 0.
• Student Question: "Can you distribute the index $n$ in an expression like $(1 + \frac{2}{n})^n$?"
  • Response: No, this is a binomial raised to a power; distributing is mathematically incorrect. It requires expansion or the exponential identity approach.
• Discussion on Class Environment: Students and the professor discussed the temperature of the room. The professor noted he prefers hot rooms to sleep but cool rooms for class to stay alert. A student mentioned using Burt's Bees under the eyes to stay awake, which "burns."
• Calculator Navigation: There was a long discussion on how to enter fractions into the TI-84. The professor admitted to being "too lazy" to find the fraction shortcut and prefers using parentheses for numerators and denominators.
• Inter-Student Exchange: One student joked about hitting their heads on the desk/wall when they fall asleep to wake up. Another mentioned the effectiveness of energy drinks.