Question

why does \[\sum_{n=1}^{\infty} \frac{ 1 }{ n }\] not converge?

Answer 1

You have to remember that it is a series and not a limit. If it were a limit then it would converge to 0, but in this case you are adding every result to the previous one, which would carry the summation off to infinity.

Answer 2

OK, I can understand that. what with n squared though? it would also keep adding positive numbers (much smaller ones?) so it should diverge?

Answer 3

Yes, it would be a p-series with n greater than 1. With 1/n, the series stays pretty consistent, only increasing by increments of 1 in the denominator. With n higher than 1, the series getting increasingly smaller much faster. Basically, the series approaches 0 FASTER than it approaches infinity. I think that would be the way to think of it. Is my series going to infinity faster or to 0 faster. Who wins the race.

Answer 4

does the 1/n one diverge towards infinity?

Answer 5

ok

Answer 6

thanks