In 5:22 the proof, From the third line to the forth line, we say that the sum from n to infinity | P.T | is convergent therefore the sum from n to infinity P.T is convergent, which uses Absolute convergence test. So we use Absolute convergence test in the process of proving Absolute convergence test, isn't this circular reasoning?
Not quite. |P.T.|=P.T., so in this case the two series are the same and there is nothing to prove. Or think of it this other way. We first "prove" the Absolute Convergent Test for positive series (there is nothing to prove) and for negative series (factor out the common -1), and then we use this to prove it for arbitrary series.
When will we learn about the Yeetus Deletus theorem?
In 5:22 the proof, From the third line to the forth line, we say that the sum from n to infinity | P.T | is convergent therefore the sum from n to infinity P.T is convergent, which uses Absolute convergence test. So we use Absolute convergence test in the process of proving Absolute convergence test, isn't this circular reasoning?
Not quite. |P.T.|=P.T., so in this case the two series are the same and there is nothing to prove. Or think of it this other way. We first "prove" the Absolute Convergent Test for positive series (there is nothing to prove) and for negative series (factor out the common -1), and then we use this to prove it for arbitrary series.
@@mat1378 Oh I see. The P.T and N.T is like axioms which is self-eivdent?
Conditionally convergent series are so counter-intuitive. It’s almost like magic