A monotonic sequence is a sequence in which the terms either strictly increase or strictly decrease. This means that the terms of the sequence either consistently get larger or consistently get smaller as the sequence progresses.

One example of a monotonic sequence is the sequence of natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … In this sequence, the terms strictly increase by 1 each time, so the sequence is monotonically increasing.

Another example of a monotonic sequence is the sequence of squares of natural numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, … In this sequence, the terms are obtained by squaring the natural numbers, so they strictly increase as the natural numbers increase. Since the terms strictly increase, this sequence is also monotonically increasing.

Monotonic sequences are important in many areas of mathematics, including calculus and analysis. In calculus, the monotonicity of a function is often used to determine whether the function has certain properties, such as differentiability or continuity. In analysis, monotonic sequences are used to define and study various mathematical concepts, such as convergence and divergence.

For example, in calculus, if a function f is monotonically increasing on an interval [a,b], then it is differentiable on that interval. This means that the derivative of f exists at every point in the interval, and it can be used to study the behavior of the function. Similarly, if a function f is monotonically decreasing on an interval [a,b], then it is also differentiable on that interval, and its derivative can be used to study its behavior.

In analysis, a sequence {x_n} is said to converge to a limit L if, for every ε>0, there exists an integer N such that |x_n – L| < ε for all n≥N. This means that the terms of the sequence get arbitrarily close to the limit L as the sequence progresses. If a sequence {x_n} is monotonically increasing and bounded above, then it is convergent. This means that the terms of the sequence get arbitrarily close to some real number L as the sequence progresses. Similarly, if a sequence {x_n} is monotonically decreasing and bounded below, then it is also convergent.

In summary, a monotonic sequence is a sequence in which the terms either strictly increase or strictly decrease. This property is important in many areas of mathematics, including calculus and analysis. Monotonic sequences are used to study the behavior of functions and to define and study various mathematical concepts, such as convergence and divergence.