i-want-the-formolla-of-taylor-and-maclorien-series-of-this-function-1-f-x-ln-x-2-f-x-sec-x-3-f-x-csc-x-4-f-x-cot-x-5-f-x-tan-x-6-f-x-sech-x

Question Number 145874 by Mrsof last updated on 09/Jul/21

i w a n t t h e f o r m o l l a o f t a y l o r a n d m a c l o r i e n

s e r i e s o f t h i s f u n c t i o n

(1) f (x) = l n (x) (2) f (x) = s e c (x)

(3) f (x) = c s c (x) (4) f (x) = c o t (x)

(5) f (x) = t a n (x) (6) f (x) = s e c h (x)

(7) f (x) = t a n h (x) (8) f (x) = c s c h (x)

(9) f (x) = c o t h (x) (10) f (x) = c o s h (x)

h o w c a n h e l p m e p l e a s e

Commented by Mrsof last updated on 09/Jul/21

? ? ? ? ? ? ?

Answered by Olaf_Thorendsen last updated on 09/Jul/21

(2)

f (x) = \sec (x) = \frac{1}{\cos x}

f (x) = \underset{n = 0}{\sum^{\infty}} \frac{E_{2 n}}{(2 n)!} x^{2 n}

(3)

f (x) = \csc (x) = \frac{1}{\sin x}

f (x) = \frac{1}{x} + \underset{n = 1}{\sum^{\infty}} ∣ B_{2 n} ∣ \frac{2 (2^{2 n - 1} - 1)}{(2 n)!} x^{2 n}, 0 <∣ x ∣< π

(4)

f (x) = \cot (x)

f (x) = \frac{1}{x} - \underset{n = 1}{\sum^{\infty}} ∣ B_{2 n} ∣ \frac{2^{2 n}}{(2 n)!} x^{2 n - 1}, 0 <∣ x ∣< π

(5)

f (x) = \tan (x)

f (x) = \underset{n = 1}{\sum^{\infty}} ∣ B_{2 n} ∣ \frac{2^{2 n} (2^{2 n} - 1)}{(2 n)!} x^{2 n - 1}, ∣ x ∣< \frac{π}{2}

(6)

f (x) = sech (x) = \frac{1}{ch (x)}

f (x) = \underset{n = 0}{\sum^{\infty}} {(- 1)}^{n} \frac{E_{2 n}}{(2 n)!} x^{2 n}

(7)

f (x) = \tanh (x)

f (x) = \underset{n = 1}{\sum^{\infty}} B_{2 n} \frac{2^{2 n} (2^{2 n} - 1)}{(2 n)!} x^{2 n - 1}, ∣ x ∣< \frac{π}{2}

(8)

f (x) = csch (x) = \frac{1}{sh (x)} = \frac{2 e^{x}}{e^{2 x} - 1}

f (x) = \frac{1}{x} - \underset{n = 1}{\sum^{\infty}} B_{2 n} \frac{2 (2^{2 n} - 1)}{(2 n)!} x^{2 n - 1}, 0 <∣ x ∣< π

(9)

f (x) = \coth (x)

f (x) = \frac{1}{x} + \underset{n = 1}{\sum^{\infty}} B_{2 n} \frac{2^{2 n}}{(2 n)!} x^{2 n - 1}, 0 <∣ x ∣< π

(10)

f (x) = \cosh (x)

f (x) = \underset{n = 0}{\sum^{\infty}} \frac{1}{(2 n)!} x^{2 n}

Commented by tabata last updated on 09/Jul/21

t h a n k y o u s i r b u t w h a t s t h e m e a n B_{2 n} a n d E_{2 n}

Commented by Olaf_Thorendsen last updated on 09/Jul/21

E_{k} = Euler numbers .

B_{k} = Bernoulli numbers .