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How to represent sequences?
$$\{a_n\}^\infty_1$$
What is the general form of a series?
$$s_n = \sum_{i=1}^{n} a_i$$
How to represent an infinite series?
$$s_\infty = \lim\limits_{n \to \infty} \sum_{i=1}^{n} a_i$$
What is a sequence?
A list of terms related by a common pattern.
What is a convergent sequence?
A sequence where $\lim_{n \to \infty} a_n$ exists and is finite.
What is a divergent sequence?
A sequence where $\lim_{n \to \infty} a_n$ does not exist or is infinite.
Define an increasing sequence.
A sequence where $\frac{a_{n+1}}{a_n} > 1$ for all n.
Define a decreasing sequence.
A sequence where $\frac{a_{n+1}}{a_n} < 1$ for all n.
What is a monotonic sequence?
A sequence that is either increasing or decreasing.
What does it mean for a sequence to be bounded above?
There exists an upper bound to the sequence.
What does it mean for a sequence to be bounded below?
There exists a lower bound to the sequence.
What is a bounded sequence?
A sequence that is both bounded above and below.
What is a series?
A sum of the terms in a sequence.
What is the $n^{th}$ partial sum?
The value of the summation of the 1st through the $n^{th}$ terms.
What is an infinite series?
A series where n=$infty$, or $s_\infty = \lim\limits_{n \to \infty} \sum_{i=1}^{n} a_i$.
What is a convergent series?
A series in which $s_\infty$ exists and is finite.
What is a divergent series?
A series in which $s_\infty$ does not exist or is infinite.
What is a telescoping series?
A series where the middle terms cancel out, leaving only the first and last terms.
What does the Sequence Convergence Theorem state?
If a sequence is both bounded and monotonic, then the sequence converges.