let-f-x-2x-1-ln-1-x-2-1-calculate-f-n-x-and-f-n-0-2-developp-f-at-integr-serie-3-calculate-0-1-f-x-dx-

Question Number 55269 by maxmathsup by imad last updated on 20/Feb/19

l e t f (x) = (2 x + 1) l n (1 - x^{2})

1) c a l c u l a t e f^{(n)} (x) a n d f^{(n)} (0)

2) d e v e l o p p f a t i n t e g r s e r i e .

3) c a l c u l a t e \int_{0}^{1} f (x) d x .

Commented by turbo msup by abdo last updated on 02/Mar/19

1) l e i b n i z g o r m u l a g i v e

f^{(n)} (x) = \sum_{k = 0}^{n} C_{n}^{k} {(2 x + 1)}^{(k)} (l n {(1 - x^{2})}^{(n - k)}

= (2 x + 1) (l n {(1 - x^{2})}^{(n)} + 2 n (l n {(1 - x^{2})}^{(n - 1)}

w e h a v e l n {(1 - x^{2})}^{(1)} = \frac{- 2 x}{1 - x^{2}} \Rightarrow

(l n {(1 - x^{2})}^{(n)} = {(- \frac{2 x}{1 - x^{2}})}^{(n - 1)}

= {(\frac{2 x}{x^{2} - 1})}^{(n - 1)} = {(\frac{1}{x + 1} + \frac{1}{x - 1})}^{(n - 1)}

= {(\frac{1}{x + 1})}^{(n - 1)} + {(\frac{1}{x - 1})}^{(n - 1)}

= \frac{{(- 1)}^{n - 1} (n - 1)!}{{(x + 1)}^{n}} + \frac{{(- 1)}^{n - 1} (n - 1)!}{{(x - 1)}^{n}}

Commented by turbo msup by abdo last updated on 02/Mar/19

a l s o {(l n (1 - x^{2}))}^{(n - 1)}

= \frac{{(- 1)}^{n - 2} (n - 2)!}{{(x + 1)}^{n - 1}} + \frac{{(- 1)}^{n - 2} (n - 2)!}{{(x - 1)}^{n - 1}} \Rightarrow

f^{(n)} (x) = (2 x + 1) {(- 1)}^{n - 1} (n - 1)! {\frac{1}{{(x + 1)}^{n}} + \frac{1}{{(x - 1)}^{n}}}

+ 2 n {(- 1)}^{n - 2} (n - 2)! {\frac{1}{{(x + 1)}^{n - 1}} + \frac{1}{{(x - 1)}^{n - 1}}}

Commented by maxmathsup by imad last updated on 02/Mar/19

\Rightarrow f^{(n)} (0) = {(- 1)}^{n - 1} (n - 1)! {1 + {(- 1)}^{n}} + 2 n {(- 1)}^{n} (n - 2)! {1 + {(- 1)}^{n - 1}}

= - {(- 1)}^{n} (n - 1)! {1 + {(- 1)}^{n}} + 2 n {(- 1)}^{n} (n - 2)! {1 - {(- 1)}^{n}} \Rightarrow

f^{(n)} (0) = - {(- 1)}^{n} (n - 1) (n - 2)! {1 + {(- 1)}^{n}} + 2 n {(- 1)}^{n} (n - 2)! {1 - {(- 1)}^{n}}

= {(- 1)}^{n} (n - 2)! {(1 - n) (1 + {(- 1)}^{n}) + 2 n {1 - {(- 1)}^{n}}}

= {(- 1)}^{n} (n - 2)! {1 + {(- 1)}^{n} - n - n {(- 1)}^{n} + 2 n - 2 n {(- 1)}^{n}}

= {(- 1)}^{n} (n - 2)! {1 + (1 - 3 n) {(- 1)}^{n} + n}

Commented by maxmathsup by imad last updated on 02/Mar/19

f^{^{'}} (x) = 2 l n (1 - x^{2}) + (2 x + 1) \frac{- 2 x}{1 - x^{2}} \Rightarrow f^{^{'}} (0) = 0 a n d f (0) = 0

f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (0)}{n!} x^{n} = \sum_{n = 2}^{\infty} \frac{{(- 1)}^{n}}{n (n - 1)} {n + 1 + (1 - 3 n) {(- 1)}^{n}} x^{n}

= \sum_{n = 2}^{\infty} \frac{(n + 1) {(- 1)}^{n} + 1 - 3 n}{n (n - 1)} x^{n}