Comprehensive Calculus II Study Notes: Sequences, Series, and the Integral Test

Converting Sequences to Functions: To determine if a sequence is increasing or decreasing, the sequence index $n$ is treated as a continuous variable $x$ . This allows for the application of Calculus I derivatives to the function $f(x)$ .
Derivative Test for Monotonicity:
- If the derivative $f'(x) > 0$ for $n \ge 1$ , the function is increasing.
- If the derivative $f'(x) < 0$ for $n \ge 1$ , the function is decreasing.
Case Study: $a_n = 3 - 2ne^{-n}$ :
- The function is represented as $f(x) = 3 - 2xe^{-x}$ .
- Using the product rule: $f'(x) = 0 - 2 [ (1)e^{-x} + x(e^{-x})(-1) ]$ .
- Simplified: $f'(x) = -2e^{-x} + 2xe^{-x} = -2e^{-x}(1 - x) = \frac{-2(1 - x)}{e^x}$ .
- Finding Critical Numbers: The derivative is zero or undefined at $x = 1$ .
- Sign Chart Analysis: Since we only care about positive values where $n \ge 1$ , we test a value like $x = 2$ .
- $f'(2) = \frac{-2(1 - 2)}{e^2} = \frac{2}{e^2}$ . Since this is positive, the function is increasing for all $n > 1$ .
- Applying Limits: $\lim_{n \to \infty} (3 - 2n e^{-n})$ . This is equivalent to $3 - \lim_{n \to \infty} \frac{2n}{e^n}$ .
- By L'HBpital's Rule (infinity over infinity form): $3 - \lim_{n \to \infty} \frac{2}{e^n} = 3 - 0 = 3$ .
Boundedness and Monotonicity:
- Because the sequence is increasing and its limit is $3$ , the sequence is bounded above by $3$ .
- The sequence is bounded below by its first term, $a_1 = 3 - \frac{2}{e}$ , because it is increasing.

Monotonicity: A shorthand term used in theorems. A sequence is monotonic if it is either entirely increasing or entirely decreasing.
- To be increasing: Each term must be larger than the previous term ( $a_{n+1} > a_n$ ).
- To be decreasing: Each term must be smaller than the previous term ( $a_{n+1} < a_n$ ).
Non-Monotonic Sequences: Sequences that alternate, such as those including a term like $(-1)^n$ , are not monotonic because they jump above and below specific values (e.g., alternating around an asymptote).
Monotonic Sequence Theorem: Every bounded monotonic sequence is convergent.
Divergence: A sequence is divergent if it is getting farther away from the x-axis or does not settle into a single finite value.

Logarithmic Limit Example ( $a_n = \ln(n + 1) - \ln(n)$ ):
- Using the Quotient Rule for logs: $a_n = \ln(\frac{n + 1}{n})$ .
- Applying the limit: $\ln(\lim_{n \to \infty} \frac{n + 1}{n}) = \ln(1) = 0$ .
- Continuity Rule: If the outside function is continuous, the limit can be brought inside: $\lim_{n \to \infty} f(g(n)) = f(\lim_{n \to \infty} g(n))$ .
Arctangent Limit Example ( $a_n = \arctan(\ln(n))$ ):
- Because Arctan and Natural Log are continuous, we bring the limit inside step-by-step.
- As $n \to \infty$ , $\ln(n) \to \infty$ .
- The limit becomes $\lim_{u \to \infty} \arctan(u)$ .
- Based on the horizontal asymptote of the arctan function, $\lim_{u \to \infty} \arctan(u) = \frac{\pi}{2}$ .

Infinite Series Definition: Adding an infinite number of terms ( $\sum_{n=1}^{\infty} a_n$ ).
Partial Sums ( $S_n$ ): Since we cannot physically add infinite terms, we look at the sequence of partial sums.
- $S_1 = a_1$
- $S_2 = a_1 + a_2$
- $S_3 = a_1 + a_2 + a_3$
- $S_n = \sum_{i=1}^{n} a_i$
Convergence of Series: A series is convergent if the sequence of partial sums has a finite limit $L$ . If the limit does not exist, the series is divergent.

Structure: $a + ar + ar^2 + ar^3 + \dots + ar^{n-1}$ .
Common Ratio ( $r$ ): A real number ratio between consecutive terms.
Partial Sum Closed Form Derivation:
- $S_n = a + ar + ar^2 + \dots + ar^{n-1}$
- $rS_n = ar + ar^2 + ar^3 + \dots + ar^n$
- Subtracting the two: $S_n - rS_n = a - ar^n$
- Factoring: $S_n(1 - r) = a(1 - r^n)$
- $S_n = \frac{a(1 - r^n)}{1 - r}$
Geometric Series Convergence Criteria:
- If $|r| < 1$ , the term $r^n \to 0$ as $n \to \infty$ . The sum converges to $\frac{a}{1 - r}$ .
- If $|r| \ge 1$ , the series diverges. Special case: if $r=1$ , the limit is undefined in the formula directly but diverges as it would be $a + a + a \dots$

Finding the Sum: Given a series with $a = 5$ and $r = \frac{5}{7}$ .
- Since $| \frac{5}{7} | < 1$ , it converges.
- Sum = $\frac{5}{1 - \frac{5}{7}} = \frac{35}{7 - 5} = \frac{35}{2} = 17.5$ .
Ratio Identification Methods:
- Method 1 (Algebraic Manipulation): Rewrite the term to fit the form $ar^{n-1}$ . For example, $5 \cdot \pi^{-n}$ can be written as $\frac{5}{\pi} \cdot (\frac{1}{\pi})^{n-1}$ .
- Method 2 (Division): Divide consecutive terms: $r = \frac{a_{n+1}}{a_n}$ .
Trigonometric Ratio Example: $\sum_{k=1}^{\infty} (\sin(100))^k$ .
- Here $r = \sin(100)$ . This is a constant value between $-1$ and $1$ (as $100$ is not an odd multiple of $\frac{\pi}{2}$ ).
- Sum = $\frac{\sin(100)}{1 - \sin(100)}$ .
Variable Geometric Series: $\sum_{n=1}^{\infty} (x + 2)^n$ .
- $r = x + 2$ . For convergence, $-1 < x + 2 < 1$ , which implies $-3 < x < -1$ .
- The first term $a = x + 2$ . The sum is $\frac{x + 2}{1 - (x + 2)} = \frac{x + 2}{-x - 1}$ .

The Divergence Test (Sanity Check): If $\lim_{n \to \infty} a_n \neq 0$ , then the series $\sum a_n$ MUST be divergent.
- Note: If the limit is zero, the series might still diverge (e.g., the Harmonic Series $\sum \frac{1}{n}$ ).
The Integral Test:
- Compare the series $\sum a_n$ to the improper integral $\int_{1}^{\infty} f(x) dx$ .
- Requirements for $f(x)$ : must be continuous, positive, and decreasing for $x \ge 1$ .
- If the integral converges, the series converges. If the integral diverges, the series diverges.
- Example Case: $\sum \frac{n^3}{n^4 + 4}$ .
  - Integral: $\int_{1}^{\infty} \frac{x^3}{x^4 + 4} dx$ .
  - Substitution: $u = x^4 + 4$ , $du = 4x^3 dx$ .
  - The integral result involves $\ln(u)$ , which goes to $\infty$ as $x \to \infty$ . Therefore, the series diverges by the integral test.
p-series Test: A series $\sum \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \le 1$ .

Question: What is the "Bologna" analogy?
- Response: A joke regarding the texture and grossness of hot dogs and wet beans/bologna, serving as a brief classroom aside.
Question: Can we find the sum using the integral test?
- Response: No. The integral test only tells us if the series converges or diverges. The integral value acts as an upper bound, not the exact sum. Only geometric series sums are being calculated at this stage.
Question: What if the limit of the terms is zero but it doesn't converge?
- Response: The Harmonic series $\sum \frac{1}{n}$ is the primary example. The terms effectively move toward zero, but not fast enough to prevent the sum from growing to infinity.
Student Confusion (Phone Issue): A student accidentally "erased all data" on their phone while trying to use a "Find My" function during a study session, leading to a discussion about device functionality and data loss.
Tutoring/Exam Schedule:
- The upcoming test covers up to Section 11.3.
- The test includes polar coordinates (area, derivatives, length), sequences (11.1), series (11.2), and the integral test (11.3).
- Tutoring is available on Wednesdays (and possibly Thursdays), with walk-in sessions on Fridays.