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Hi ii) follow the hint which is given... iii) : Are you allowed to use Stirling's approximation ? (for , ) : Think about the alternating series test.
Originally Posted by szpengchao The second one is asking for First you do rato test Since simplifying we would have Now we can see that Converges since and Now for the other test use stirlings approximation I showed you the first part in case you needed the interval of convergence
Originally Posted by flyingsquirrel What link is there between " for " and " diverges" ? Yeah I saw that...I changed it before you posted....I meant convergent by alternating series test
please check if my proof for the first part is right
Originally Posted by szpengchao please check if my proof for the first part is right where did that come from?
the first step is wallis product
Originally Posted by szpengchao Letting L equal that limit we get Take the natural logs of both sides and note taht
Originally Posted by szpengchao please check if my proof for the first part is right If you're allowed to use the Wallis product, (Is it in the exercise ?) it's OK.
just want to check if there are some wrong steps... ok...cheers! so for the last part, you did ratio test and try to find the radius of convergence, right? then the result comes straight from it?
Originally Posted by szpengchao just want to check if there are some wrong steps... ok...cheers! so for the last part, you did ratio test and try to find the radius of convergence, right? then the result comes straight from it? Yes I found the radius of convergence and showed that is convergent Then we msut check the endpoints...through the alternating series test we can see that makes the series converge and I leave it up to you to show that the other value makes the series ____
i have no idea to show when x=1/4 please give some hint
Originally Posted by szpengchao i have no idea to show when x=1/4 Originally Posted by flyingsquirrel : Are you allowed to use Stirling's approximation ? (for , ) ...
yeah...i got as n goes to infinity, that is equivalent to 1/sqrt( pi * n) so, what's next?
As for , and diverges, (p-series with ) the series diverges too. It's a theorem but I don't know how it is called in English.
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