when taking the summation of from 1 to infinity,
what is the difference between saying that it diverges or converges to 0?
I am kind of puzzled by this because, for example the summation from 1 to infinity of , what does this do?
does it converge to 0 after using l'hopitals rule, or does it follow the harmonic series rule and diverge?
you say is divergent because of the p series, but isnt the series for it being greater than or less than 1? if its a big proof just reply with because I said so, ill understand.
but, so are you saying that if a question or function reduces through some kind of comparison or test, like the integral or nth term test, then if it does reduce to it would actually converge instead of diverge?
oh and im sorry the equation is
The difference between a series diverging or converging to zero is a big difference. When a series diverges, that means the series has no sum. If a series converges to zero, that means that the sum of the series exists and is equal to zero.
The series diverges; although L'Hospital's Rule can tell you that the limit of the terms is zero, this does not imply that the series converges. If the limit of the terms (as n approaches infinity) is not zero, then you can conclude the series diverges.
You show that diverges by comparison with the harmonic series.