Direct Comparison vs. Limit Comparison tests

[Tom29]

I'm having trouble distinguishing between the two and when to use which when figuring out if a series converges or diverges. Can someone tell me, in simple english (because math books and websites can't) when to use which test?

Also, when comparing, how do I know what to compare it to? For example, with the limit as n goes to infinity for n / n^3 + 1 my book says to compare it to 1/n^3. But as n goes to infinity for n / n^3 - 1 my book says compare it to 1/n^2. Does it matter? I'm so confused.

 

[pka]

I'm having trouble distinguishing between the two and when to use which when figuring out if a series converges or diverges. Can someone tell me, in simple english (because math books and websites can't) when to use which test?

The answer to that question is a simple no, because it is not simple, in English or any other language. It really takes your very intense study and your doing maybe as many as one hundred problems. I tell all my students: “COPY, COPY”. Copy out all the worked examples in your text. Then go get a different text and copy the worked examples in it.

 

[Tom29]

I guess my issue is that I really don't get what the difference is. It's the only trouble I have regarding convergence tests. Integral, Ratio, Root, Alternating, Power Series, it all comes to me quickly. But Comparison, I know the concept, but it just seems like it's arbitrary what to compare it to, and comparing it to 1/n or 1/n^2 will both yield seeming correct results even when one diverges and the other converges. :?

 

[pka]

It seems to from what you have written that you don’t get the basic concept.
Do you understand that series are about sums of finite collections of numbers?
That we look at sequences of partial sum?
If for each n, \(\displaystyle a_n \le \frac{1}{{n^2 }}\) then the partial sums of the \(\displaystyle a_n\) will be \(\displaystyle \le\) the partial sums of the \(\displaystyle \frac{1}{{n^2 }}\). Thus by dominance the \(\displaystyle a_n\) series converges.

 

[stapel]

Comparison [Test]...it just seems like it's arbitrary what to compare it to....

That is probably because, to a certain extent, the choice is arbitrary. Some comparisons will work; some won't. Some will easily; some will work only with some difficulty. Some will work messily; some will be very clever.

Not everything is going to be a simple formula or algorithm. Sometimes you're going to have to be a little creative, and take some time to try different things. This will generally become more true, the further you get in mathematics.

Eliz.