Question

Does the following series converge or diverge? Why?
Summation k^(1/3)/sqrt(k^3+k^2+1) from k=1 to infinity

Answer 1

It is converging, the answer is fairly simple.

We split the equation up in two parts, the sum, and the function inside.
We now see how the function behave under different K.
As small K there is no problem, but as K increases the result under the fraction line goes to infinity, in a rate much higher then the result over the fraction line, the result of the function as k gets bigger becomes smaller and smaller,
When all this is summed up, you get some value from the low values of k, will the results from the function with high k becomes 0

Answer 2

Hansen, that's not necessarily true - the mere fact that the terms go to zero does not imply that the summation converges. Take the harmonic series - \[\sum_{k=1}^{\infty} 1/k\], which does not converge even though the terms go to zero.

Answer 3

Yeah, but if you take \[\sum_{1}^{\infty}{ {1}\over{k^2}}\] Then this converges, because you start getting \[{1}\over{\infty^2}\]