Definition of sequence in Calculus.

A sequence is an ordered list of objects (or events).
Also, the sequence $(1, 1, 2, 3, 5, 8)$ , which contains the number $1$ at two different positions, is a valid sequence.
Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers $(2, 4, 6, \cdots)$ .
Finite sequences are sometimes known as strings or words, and infinite sequences as streams.
The empty sequence $( \quad )$ is included in most notions of sequence, but may be excluded depending on the context.

A series is the sum of the terms of a sequence.
Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.
The sequence of partial sums ${S_k}$ associated to a series $\sum_{n=0}^\infty a_n$ is defined for each k as the sum of the sequence ${a_n}$ from $a_0$ to $a_k$ :
Infinite sequences and series can either converge or diverge.
An infinite sequence of real numbers shown in blue dots.

A series is the sum of the terms of a sequence.
A series is, informally speaking, the sum of the terms of a sequence.
Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.
Given an infinite sequence of numbers $\{ a_n \}$ , a series is informally the result of adding all those terms together: $a_1 + a_2 + a_3 + \cdots$ .
The sequence of partial sums $\{S_k\}$ associated to a series $\sum_{n=0}^\infty a_n$ is defined for each k as the sum of the sequence $\{a_n\}$ from $a_0$ to $a_k$ :

Like any series, an alternating series converges if and only if the associated sequence of partial sums converges.
Proof: Suppose the sequence $a_n$ converges to $0$ and is monotone decreasing.
(The sequence $\{ S_m \}$ is said to form a Cauchy sequence, meaning that elements of the sequence become arbitrarily close to each other as the sequence progresses.)

For a sequence $\{a_n\}$ , where $a_n$ is a non-negative real number for every $n$ , the sum $\sum_{n=0}^{\infty}a_n$ can either converge or diverge to $\infty$ .
For a sequence $\{a_n\}$ , where $a_n$ is a non-negative real number for every $n$ , the sequence of partial sums
Therefore, it follows that a series $\sum_{n=0}^{\infty} a_n$ with non-negative terms converges if and only if the sequence $S_k$ of partial sums is bounded.

In this example, the ratio of adjacent terms in the blue sequence converges to $L=\frac{1}{2}$ .
Then the blue sequence is dominated by the red sequence for all $n \geq 2$ .
The red sequence converges, so the blue sequence does as well.

A transcendental function is a function that "transcends" algebra in the sense that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, power, and root extraction.
Identify a transcendental function as one that cannot be expressed as the finite sequence of an algebraic operation

Since the sum of the sequence $\frac{1}{n}$ $\left ( \text{i.e., }\sum {\frac{1}{n}}\right)$ diverges, the limit convergence test tells that the original series (with $\frac{n+1}{2n^2}$ ) also diverges.

The above examples involving the harmonic series raise the question of whether there are monotone sequences such that $f(n)$ decreases to $0$ faster than $\frac{1}{n}$ but slower than $\frac{1}{n^{1 + \varepsilon}}$ in the sense that:
Once such a sequence is found, a similar question can be asked of $f(n)$ taking the role of $\frac{1}{n}$ oand so on.

However, $f$ itself is not continuous as can be seen by considering the sequence $f \left(\frac{1}{n},\frac{1}{n} \right)$ (for natural $n$ ), which should converge to $\displaystyle{f (0,0) = 0}$ if $f$ is continuous.

sequence