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):

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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):

Mathematics
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\[\sum_{1}^{\infty}\]\[k ^{-1/3}a _{k}^{1/2}\]
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 :)

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