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):

Question

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 justified
by either a proof if true or an example is false):

anonymous · Answer

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

anonymous · Answer

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 :)