## What is a sequence of a real number?

Definition 1.1 A sequence of real numbers is a function from the set N of natural numbers to the set R of real numbers. If f : N → R is a sequence, and if an = f(n) for n ∈ N, then we write the sequence f as (an) or (a1,a2,…). A sequence of real numbers is also called a real sequence.

**How do you find the boundedness of a sequence?**

A sequence is bounded if it is bounded above and below, that is to say, if there is a number, k, less than or equal to all the terms of sequence and another number, K’, greater than or equal to all the terms of the sequence. Therefore, all the terms in the sequence are between k and K’.

### How do you tell if a series converges for all real numbers?

If there exists a real number R>0 such that the series converges for |x−a|R, then R is the radius of convergence. If the series converges only at x=a, we say the radius of convergence is R=0. If the series converges for all real numbers x, we say the radius of convergence is R=∞ (Figure 10.1.

**What is a real sequence?**

Definition: (Real sequence) : A real sequence or a sequence of real numbers is defined as a function from N, the set of natural number to R, the set of real numbers, i.e. f : N → R. It is denoted by < f1,f2., fn, > or < fn > or (fn).

#### Does sequence (- 1 n converge?

They never converge to any value. So, we say that the infinite series diverges.

**How do you know if a series is convergent or divergent?**

convergeIf a series has a limit, and the limit exists, the series converges. divergentIf a series does not have a limit, or the limit is infinity, then the series is divergent. divergesIf a series does not have a limit, or the limit is infinity, then the series diverges.

## How do you know if a sequence is convergent or divergent?

Precise Definition of Limit If limn→∞an lim n → ∞ exists and is finite we say that the sequence is convergent. If limn→∞an lim n → ∞ doesn’t exist or is infinite we say the sequence diverges.

**What are real and tonal sequences?**

In a melody, a real sequence is a sequence where the subsequent segments are exact transpositions of the first segment, while a tonal sequence is a sequence where the subsequent segments are diatonic transpositions of the first.