Sequences and Series Convergence/Divergence

Sequences and Series: Convergence and Divergence

• The central question is whether a sequence or series converges or diverges, and how to prove it.
• Convergence/divergence tests will be used, which are like mini-proofs.

Terms in a Sequence

• Each number in a sequence is called a term.
• If a term is represented as $\frac{5n + 1}{n^2}$, the first term is found by substituting $n = 1$, resulting in $\frac{5(1) + 1}{(1)^2} = 6$.

Determining Convergence

• To determine if a sequence converges or diverges, analyze the limit as $n$ approaches infinity.
• Example: For the sequence $\frac{5n + 1}{n^2}$, as $n \to \infty$, the sequence approaches 0 because the denominator grows faster than the numerator.
• If a sequence approaches 0 as $n \to \infty$, it converges.

Sequence vs. Series

• Sequence: A list of terms.
• Series: The sum of all the terms in a sequence.
• Currently, the focus is on sequences and determining whether they converge or diverge.

Proving Convergence or Divergence Using Limits

• To prove convergence or divergence, evaluate the limit as $n$ approaches infinity.
• Example: $\lim_{n \to \infty} \frac{6n^2}{5^n}$
• Cannot directly apply shortcuts because it's not a polynomial over a polynomial.

Rates of Growth

• Different functions grow at different rates as $n$ approaches infinity.
• Polynomial: Variable raised to a power (e.g., $x^2, x^3$).
• Exponential: Constant raised to a variable (e.g., $5^n$).
• Factorial: Denoted by an exclamation point (e.g., $n!$), represents the product of all positive integers up to $n$.

Factorial Growth

• $n! = n \times (n-1) \times (n-2) \times … \times 1$
• Example: $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$

Comparing Growth Rates

• Growth Rate (Slowest to Fastest): Polynomial < Exponential < Factorial
• In the example $\frac{6n^2}{5^n}$, the denominator (exponential) grows faster than the numerator (polynomial).
• Therefore, the limit as $n \to \infty$ is 0, and the sequence converges.

Example: Divergent Sequence

• Consider a sequence where the numerator grows faster than the denominator.
• The limit as $n \to \infty$ would be infinity, indicating divergence.

Logarithmic Growth

• Logarithmic growth (e.g., $\ln(x)$) is slower than polynomial growth.

Example: Sequence with Cosine

• Consider the sequence $\frac{\cos(n)}{n^2}$.
• $\cos(n)$ oscillates between -1 and 1.
• The denominator, $n^2$, grows to infinity.
• Thus, the limit as $n \to \infty$ is 0 because a bounded number divided by an infinitely large number approaches zero; hence the sequence converges.

Zero as a Number

• Zero is an even number because it is divisible by 2 without leaving a remainder (i.e., $0 / 2 = 0$).
• To be even, a number must have a factor of 2.
• $0 = 2 \times 0$

Last Example: Exponential Comparison

• Analyze $\frac{5^n}{4^n}$.
• Both numerator and denominator are exponential, but the numerator grows faster.
• Therefore, the limit as $n \to \infty$ is infinity, and the sequence diverges.

Connection to Geometric Sequences

• The expression $\frac{5^n}{4^n}$ can be rewritten as $(\frac{5}{4})^n$, which is in the form of a geometric sequence.
• Geometric sequence converges when the absolute value of $r$ (the common ratio) is less than 1 and diverges when it is greater than 1.
• In this case, |\frac{5}{4}| > 1, so it diverges.

Upcoming Topics

• Partial sums will be discussed.