This is given as the sum of n numbers divided by n.

This is given as the product of n numbers to the power of 1/n.

|x + y| ≤ |x| + |y|

∀ n, aₙ < aₙ₊₁

∀ n, aₙ ≤ aₙ₊₁

∀ n, aₙ > aₙ₊₁

∀ n, aₙ ≥ aₙ₊₁

Either increasing or decreasing or both.

Neither increasing nor decreasing.

∃ U s.t. ∀ n, aₙ ≤ U

∃ L s.t. ∀ n, aₙ ≥ L

Both bounded above and bounded below.

∀ C > 0 ∃ N s.t. aₙ > C ∀ n > N

∀ C < 0 ∃ N s.t. aₙ < C ∀ n > N

∀ ε > 0 ∃ N s.t. |aₙ| < ε ∀ n > N

∀ ε > 0 ∃ N s.t. |aₙ - a| < ε ∀ n > N

∃ k s.t. aₙ₊ₖ satisfies.

aₙₖ s.t. nₖ is natural and strictly increasing.

A real number that can be written as p/q for integers p and q, where q is nonzero.

A real number that cannot be written as p/q for integers p and q.

An upper bound which is less than any other upper bound.

A lower bound which is greater than any other lower bound.

∀ ε > 0 ∃ N s.t. |aₙ - aₘ| < ε ∀ m,n > N

A series whose partial sums converge.

A series whose partial sums don't converge.

A series whose partial sums tend to infinity.

A series whose partial sums tend to minus infinity.

A series ∑aₙ such that ∑|aₙ| is convergent.

A sequence bₙ of aₙ such that there is a bijection between the sets of terms of the sequences.

A convergent series ∑aₙ such that ∑|aₙ| is not convergent.

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