plzz solve thiss question
Answers
Heya....
Here is your answer...
sequence of real numbers, a1, a2, . . . , an, . . . will have a finite limit value or will be convergent if for no matter how small a positive number e we take there exists a term an such that the distance between that term and every term further in the sequence is smaller than e, that is, by moving further in the sequence the difference between any two terms gets smaller and smaller.
As an + r, where r = 1, 2, 3, . . . denotes any term that follows an, then| an + r - an | < e for all n > n0(e), r = 1, 2, 3, . . .
shows the condition for the convergence of a sequence.
If a sequence {an} of real numbers (or points on the real line) the distances between which tend to zero as their indices tend to infinity, then {an} is a Cauchy sequence.
Therefore, if a sequence {an} is convergent, then {an} is a Cauchy sequence.
The Cauchy criterion or general principle of convergence example
The following example shows us the nature of that condition.
Example: We know that the sequence 0.3, 0.33, 0.333, . . . converges to the number 1/3 that is, 1/3 = 0.33333 . . . . Let write the rule for the nth term,
If we go along the sequence far enough, say to the 100th term, i.e., the term with a hundred 3's in the fractional part, then the difference between that term and every next term is equal to the decimal fraction with the fractional part that consists of a hundred 0's followed by 3's on the lower decimal places, starting from the 101st decimal place. That is,
Therefore, the absolute value of the difference falls under
Then, if we go further along the sequence and for example calculate the distance between the 100000thterm
and the following terms, the distance will be smaller than
Hence, since we can make the left side of the inequality | an + r - an | < e as small as we wish by choosing n large enough, then all terms that follow an (denoted an + r, r = 1, 2, 3, . . . ), infinitely many of them, lie in the interval of the length 2e symmetrically around the point an. Outside of that interval there is only a finite number of terms.
Thanks....!!!
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Sorry baby 'wink'