sequence determining convergence of series
Theorem. Let be any series of real . If the positive numbers are such that
then the series converges simultaneously with the series
Proof. In the case that the limit (1) is positive, the supposition implies that there is an integer such that
and since the series and converge simultaneously with the series , the comparison test guarantees that the same concerns the given series
The case where (1) is negative, whence we have
may be handled as above.
Note. For the case , see the limit comparison test.
|Title||sequence determining convergence of series|
|Date of creation||2013-03-22 19:06:54|
|Last modified on||2013-03-22 19:06:54|
|Last modified by||pahio (2872)|