Sequences and Infinite Series Free Online FlashCards

What is a sequence?

A sequence is an ordered list of numbers, written as {a_n} = a_1, a_2, a_3, …, where a_n is the nth term.

How do explicit and recursive sequence definitions differ?

An explicit formula gives a_n directly, while a recursive definition supplies initial terms and a rule for obtaining later terms.

When does a sequence converge?

A sequence converges if its terms approach a finite limit L as n approaches infinity; otherwise, it diverges. The sequence (-1)^n diverges because it alternates between 1 and -1.

What does the Monotone Convergence Theorem state?

An increasing sequence bounded above, or a decreasing sequence bounded below, must converge.

How is an infinite series defined?

An infinite series is defined through its partial sums S_N = Σ_{n=1}^N a_n. The series converges to S exactly when the partial sums approach S.

What does the nth-Term Test prove?

If lim a_n is nonzero or does not exist, then Σa_n diverges. However, a_n tending to zero alone cannot prove convergence; the harmonic series is a counterexample.

When does a geometric series converge, and what is its sum?

For |r| < 1, Σ_{n=0}^∞ ar^n = a/(1-r). If |r| ≥ 1, the geometric series diverges.

How do you evaluate a telescoping series?

Expand finite partial sums to expose cancellation. For example, 1/[n(n+1)] = 1/n - 1/(n+1), so Σ_{n=1}^∞ 1/[n(n+1)] = 1.

What is the convergence rule for a p-series?

The p-series Σ1/n^p converges when p > 1 and diverges when p ≤ 1. Thus, the harmonic series Σ1/n diverges.

What are the valid directions of Direct Comparison?

If 0 ≤ a_n ≤ b_n eventually and Σb_n converges, then Σa_n converges. Conversely, if Σa_n diverges, then Σb_n diverges.

What does the Limit Comparison Test establish?

For positive terms, if lim(a_n/b_n) = L with 0 < L < ∞, then Σa_n and Σb_n have the same convergence or divergence behavior.

When does the Integral Test apply?

If f(x) is positive, continuous, and decreasing for large x, with f(n) = a_n, then Σa_n and ∫f(x) dx from the corresponding starting point to infinity either both converge or both diverge.