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A series of positive terms is convergent if from and after some fixed term , where r is a fixed number. The series is divergent if from and after some fixed term.
D’ Alembert’s Test is also known as ratio test of convergency of a series.
Definitions for Generally Interested Readers
(Definition 1) An infinite series i.e. is said to be convergent if , the sum of its first terms, tends to a finite limit as n tends to infinity.
We call the sum of the series, and write .
Thus an infinite series converges to a sum S, if for any given positive number , however small, there exists a positive integer such that
for all .
If as , the series is said to be divergent.
Thus, is said to be divergent if for every given positive number , however large, there exists a positive integer such that for all .
If does not tends to a finite limit, or to plus or minus infinity, the series is called Oscillatory
Let a series be . We assume that the above inequalities are true.
- From the first part of the statement:
, ……… where r <1.
Since r<1, therefore as
therefore =k say, where k is a fixed number.
Therefore is convergent.
- Since, then, , …….
Therefore and so on.
Therefore > . By taking n sufficiently large, we see that can be made greater than any fixed quantity.
Hence the series is divergent.
- When , the test fails.
Another form of the test–
The series of positive terms is convergent if >1 and divergent if <1.
One should use this form of the test in the practical applications.
Verify whether the infinite series is convergent or divergent.
We have and
Hence, when 1/x >1 , i.e., x <1, the series is convergent and when x >1 the series is divergent.
Take Now , a non-zero finite quantity.
But is convergent.
Hence, is also Convergent.