[微積] binomial series

[silentchaos (眷戀)]

f(x)=arcsinx

by some integral method

we get

f(x)=x+(1/6)x^3+(3/40)x^6+.....

by the convergence interval of binomial series

abs x<1


it is said x=+1,-1 f(+1,-1)=arcsin (+1,-1) exist

so it is convergent.

but for 1/(1-x^2)^(1/2) x=-1, -1  approches infinity

Just want to ask for endpoint of arcsin x x=1 x=-1

are they convergent?

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◆ From: 140.119.129.126
→ CaptainH    :為什麼硬要用英文寫
推 suhorng     :arcsin的limit存在, power series的不存在
推 suhorng     :樓上推錯 存在的, 這是Abel's Thm
→ silentchaos :謝謝s大 每次都給我很完整的解答
推 jacky7987   :簡單的說就是級數帶進去存在的話 極限就會存在
→ jacky7987   :(上面說的是邊界
推 jacky7987   :反過來是錯的,也就是極限存在不代表級數收斂
→ jacky7987   :加上條件讓他對的是Tauber's thm
→ suhorng     :十分抱歉qq
推 jacky7987   :我也常記反的說(攤手
→ sneak       : 加上條件讓他對的是Ta https://muxiv.com
→ sneak       : 為什麼硬要用英文寫 https://daxiv.com