Question

Determine whether the series converges or diverges

Answer 1

\[1+\frac{ 1 }{ 2\sqrt{2} }+\frac{ 1 }{ 3\sqrt{3} }+\frac{ 1 }{ 4\sqrt{4} }+\frac{ 1 }{ 5\sqrt{5} }\]

Answer 2

+......

Answer 3

how can i tell?

Answer 4

oops that was wrong sorry, the short answer is YES

Answer 5

\[\sum_{k=1}^{\infty}(\frac{1}{k^{\frac{3}{2}}})\]

Answer 6

i can tell because the degree of the denominator is \(\frac{3}{2}\) and the degree of the numerator is \(0\) and as long as the differences in degrees is more than 1, it converges

Answer 7

you probably have to use something like the integral test

Answer 8

\[\int_1^{\infty}\frac{1}{x^{\frac{3}{2}}}dx\] converges, so does the sum above

Answer 9

ah yess i see. why is it x^3/2 though?

Answer 10

without using the integral test (the one i just wrote) i only know the eyeball test, check the degrees

Answer 11

\[x\sqrt{x}=x^{\frac{3}{2}}\] right?

Answer 12

of right because you add the exponents lol silly me

Answer 13

so now i set this to the lim t-> inf.

Answer 14

yeah and as you can see, if the degree in the denominator is greater that one more than the degree in the numerator, it will converge

Answer 15

ah yes i see!

Answer 16

if it is exactly one more than when you integrate you get the log, so it will diverge

Answer 17

and of course if it is less than one more you have no hope
\[\int\frac{1}{\sqrt{x}}dx=2\sqrt{x}\] for example

Answer 18

no hope lol. i have no hope anyways when it comes to math.

Answer 19

somehow i don't believe that

Answer 20

haha yah well its true! lol sorry openstudy stopped working for me last night. thanks for the help!