## anonymous 5 years ago Suppose that summand from k=1 to infinite of asubk is a convergent series with positive terms. Does the following series necessarily converge (answer must be justified by either a proof if true or an example is false):

1. anonymous

$\sum_{1}^{\infty}$$k ^{-1/3}a _{k}^{1/2}$

2. anonymous

Let's write down the function in a more decent form ^_^:$=\sqrt{ak}/\sqrt[3]{k}$ now if we want to calculate the speed of both we'll write it down as follows: $(ak)^1/2 > (k)^1/3$ we know that 1/2 is > 1/3 , so we notice that the upper part is larger and much faster. So, the series diverge, since the upper part is alot faster than the lower part, it goes to infinity. Answer: the following series dosn't necessarily converge. ^_^ Hope you understood what I wrote , and please correct me if I'm wrong :)