What is the best pattern for taylor's series for arctg x , around x = 0, found with integration or differential of geometric series?

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We know that `(d(arctgx))/(dx)=1/(1+x^2)`

Hence, the best pattern for Taylor's series for `arctg x` , around `x = 0` , can be found from the series for `1/(1+x^2)` .

So,  `(d(arctgx))/(dx)` =`1/(1+x^2)` =`1-x^2+x^4-x^6+x^8` `-` .......for` -1ltxlt1`

Integrating term by term gives:

`arctgx` =`int 1/(1+x^2)`

`=C+x-x^3/3+x^5/5-x^7/7+x^9/9` ..............for `-1ltxlt1` , where `C ` is the constant of integration.

Since, `arc tg0=0` we have `C=0` , so,

`arctgx=x-x^3/3+x^5/5-x^7/7+x^9/9` ......for `-1ltxlt1` .


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