proof of limit comparison test
Suppose where can be a non negative real number or .
By definition, for finite, this means that for every there is a natural number such that for all ,
To make matters more concrete choose and assume and finite.
, for all .
If converges, so will and thus by the comparison test, will also be convergent.
For the reverse result, consider , since if is finite so will , applying the previous result we can say that if converges so will
Consider the case , clearly since both and are positive, this means that for all there exists such that for all , .
Considering we get the exact formulation of the comparison test, so if converges so will .
For the case just apply the result to to conclude that if converges so will
|Title||proof of limit comparison test|
|Date of creation||2013-03-22 15:35:54|
|Last modified on||2013-03-22 15:35:54|
|Last modified by||cvalente (11260)|