Question

Determine the convergence or divergence of the series:

Answer 1

\[\sum_{n=1}^{\infty}\frac{ \ln(n) }{ n+1 }\]

I think my main question is more in why some things are acceptable as comparions and why some are not. Like, my first guess for a comparison was \[\frac{ \ln(n) }{ n }\]You can show that ln(n)/n converges and then try to use it as a comparison, but apparently this is wrong to do. So how do you determine a correct comparison then? Can you have multiple comparisons that are logical and work or is there or ever going to be one main one, not including constant multiples of?

Answer 2

comparisons only work if you know that the comparison function is bigger or smaller than the stated function. If a bigger function function converges, then so does everything under it. If a smaller function diverges, then so does everything above it.

Answer 3

@amistre64

Is a function required to be similar, or can it just be something we know is always greater than or less than the function we want to determine the behavior of?

Answer 4

similar is a vague term.