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Sequence ,convergent sequence and divergent sequence

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Sequence

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In simple words, sequence is enumerated collection of objects in which repetitions are allowed.

In field of real analysis , a sequence is defined as :-

If N is a set of natural number and X be any set,then a function f: N→X is called a sequence.
If X=R or subset of R (where R is set of real number), then f is called a real sequence and if X=C or subset of C (where C is a set of complex number) then f is called a complex sequence.

A sequence generally denoted by { }, by putting values in { }.

Examples of sequence:-

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Convergent Sequence

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Convergent is an important property of sequence.A sequence is converge to a particular value known as limit of the sequence.

In field of real analysis, the convergent sequence is defined as:-

A sequence is said to converge to limit l , if given ε>0 , however small, there exists a positive integer m (depending upon ε) such that:-

|a_n -l |<ε for all n≥m

Problem on Convergent Sequence

Problem:-1) Show that the sequence {n+1/n} converges to 1.

Solution:- Let {a_n}={n+1/n}

Now consider given ε>0, however small, we choose a natural number m such that:-

m>1/ε or 1/m <1/ε

Therefore, for all n≥m or 1/n ≤1/m we have:-

|a_n -1| = |n+1-n/n| = 1/n but 1/n≤1/m <ε

Therefore, |a_n -1|<ε for all n≥m

⇒ limit _n→∞a_n =1

Hence sequence {n+1/n} converges to limit 1.

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Divergent Sequence

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A sequence is said to be divergent, if it is not convergent.This means a divergent sequence doesn't converge to a finite value of limit. The value of limit of divergent sequence is infinite (+∞ to -∞)

In field of real analysis:-

• A sequence {a_n} is said diverge to +∞ if k>0 however large, there exists a positive integer m(depending upon k) such that :-

a _n > k for all n ≥ m

• A sequence {a_n} is said diverge to -∞ if given k>0, however large, there exists a positive integer m(depending upon k) such that :-

a _n <-k for all n ≥ m

Problem on divergent sequence

Problem:-2) Show that the sequence {n² + 3n} diverges to +∞ .

Solution:- Let given sequence is {a_n} where a_n = n² + 3n

Let k>0 however large--

Now a_n > k

if n² + 3n > k

if n² > k-3n

if n > √k - 3n

Now let a positive integer m just greater than √k - 3n

⇒ a_n > k for all n≥m

Therefore {n² + 3n} diverges to +∞ .

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Citations:-

• (https://pixabay.com/en/geometry-mathematics-volume-surface-1044090/)(only background of image 1)..

• (https://en.wikipedia.org/wiki/Sequence) ( only simple definition of sequence is used)..

• (https://www.cliffsnotes.com/study-guides/algebra/algebra-ii/sequences-and-series/definition-and-examples-of-sequences) ( only first image of site is used as in example of sequence)

• (https://en.wikipedia.org/wiki/Limit_of_a_sequence) ( simple definitions for convergent and divergent sequences are modified from here)..

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Thanks for reading!

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