June 29, 2026 - Calculus 2 - Properties and Convergence of Sequences and Series (Concise)

Limits of sequences are often determined by the speed at which terms approach infinity.
Indeterminate forms such as $\frac{\infty}{\infty}$ can be solved using derivatives ("locals").
For limits involving powers, natural logarithms ( $\ln$ ) are used to handle the exponent. If $\ln(y) = L$ , then the limit is $e^L$ .
If the denominator grows while the numerator remains constant or grows slower, the limit is $0$ .

Bounded Above: A sequence is bounded above if there is a number $M$ such that $a_n \leq M$ for all $n$ .
Bounded Below: A sequence is bounded below if there is a number such that $a_n \geq M$ for all $n$ .
Bounded Sequence: A sequence that is bounded both above and below. If it is not, it is unbounded.
Monotonicity:
- Increasing: Each term is greater than or equal to the previous term ( $a_n \leq a_{n+1}$ ).
- Decreasing: Each term is less than or equal to the previous term ( $a_n \geq a_{n+1}$ ).
- Eventually Monotonic: A sequence that becomes increasing or decreasing after a certain index $n_0$ .
- Monotone/Monotonic: A sequence that is either eventually increasing or decreasing.

Convergence Theorem: If a sequence converges, then it is bounded.
Monotone Convergence Theorem: If a sequence is bounded and monotone, then it converges.
A bounded sequence is not guaranteed to converge (e.g., $1, 0, 1, 0 \dots$ is bounded but does not converge).

Factorial Definition: $n!$ is the product of all positive integers less than or equal to $n$ . $0!$ is defined as $1$ .
Growth Comparison: Factorials eventually grow faster than exponential functions.
Example: The sequence $\frac{4^n}{n!}$ is eventually decreasing and bounded below by $0$ , meaning it converges.

Series: An operation adding terms of a sequence together: $\sum_{i=1}^{n} a_i$ .
Infinite Series: A series that continues forever ( $\sum_{i=1}^{\infty} a_i$ ).
Partial Sums ( $s_k$ ): The sum of the first $k$ terms.
Convergence of Series: An infinite series converges if the sequence of its partial sums converges to a real number. If the partial sums do not reach a real number limit, the series diverges.

Geometric Series: A series with a common ratio $r$ .
- Converges if the ratio is between $-1$ and $1$ (|r| < 1).
- Sum formula: $S = \frac{a}{1 - r}$ , where $a$ is the first term.
Harmonic Series: The series $\sum \frac{1}{n}$ .
- It diverges towards infinity, though it grows very slowly.

Linearity: If two series converge, their sum or difference also converges to the sum or difference of their individual limits.
Constant Multiple: Multiplying a convergent series by a constant yields a convergent series equal to the constant times the original sum.
Index Shifting: To change a starting index while maintaining the same sum:
- If the index is increased, the variable $n$ within the formula must be decreased by the same amount.
- If the index is decreased, the variable $n$ must be increased.

Student: Isn't there supposed to be a $2$ if I was taking an anti-derivative?
In-Transcript Response: Those cancel; we are not in an indeterminate form anymore.
Student: What does $n!$ mean?
In-Transcript Response: It is the product of all numbers less than $n$ ( $1 \times 2 \times 3 \times 4 \times 5$ for $5!$ ).
Student: How did you go from $1 - \frac{1}{2^n}$ into $\frac{2 - 1}{2^n}$ ?
In-Transcript Response: By subtracting exponents. $n + 1 - n$ leaves $2^1$ , so the $2$ sticks around.
Student: Could you also rewrite your index to have the same power?
In-Transcript Response: You could, but the index itself would change. It is often easier to let one term stay outside the sequence.