Question

I am about to take the test on infinite series in calc II and I am a little confused because all the answers in my solutions manual keep applying l'hopital to infinite limits when determining convergence or divergence of a series. I thought you had to set the sum equal to it's associated function f(x)dx to perform l'hopital's rule. Why do these solutions keep applying l'hopital's to n terms? My instructor won't let us fudge that concept, so the process has to be correct. Which way is correct?

Answer 1

do you have the original question?

Answer 2

It is more of a general observation. For instance \[\sum_{1}^{\infty} n/(n+1)\]
Some of the solutions I see solve the limit, which is simplest, but others solve it in the form \[\lim_{n \rightarrow \infty} n/(n+1)\], and apply l'hopitals until they get the limit, if it exists as if this infinite limit is a plain old function of x. I think I am just seeing a lot of laziness in these solutions manuals because I was taught that you can't apply l'hopitals to the limit of an infinite series. I thought you must first set this series equal to f(x) = x/(x+1) to apply l'hopitals. Otherwise the only thing I see that you could correctly do is to divide out the n in the numerator by multiplying by 1/n/1/n to get a limit of 1 which fails the nth term test.

Answer 3

So what I am asking is whether you can actually just apply l'hopital to the infinite limit when you get an indeterminate form, or do you have to have set it equal to a function, and take the limit of the function as x approaches infinity to be able to apply l'hopital's rule?