Let-f-x-be-a-real-valuedfunction-x-R-Determine-the-following-sums-1-S-1-n-r-1-n-tan-1-rf-x-2-S-2-n-r-1-n-sin-1-rf-x-f-x-1-n-3-S-3-n-r-1-n-cos-1-

Question Number 1437 by 112358 last updated on 03/Aug/15

L e t f (x) b e a r e a l v a l u e d f u n c t i o n

\forall x \in R . D e t e r m i n e t h e f o l l o w i n g

s u m s :

(1) S_{1} (n) = \underset{r = 1}{\sum^{n}} {t a n}^{- 1} (r f (x))

(2) S_{2} (n) = \underset{r = 1}{\sum^{n}} {s i n}^{- 1} (r f (x)) {∣ f (x) ∣⩽ 1 / n}

(3) S_{3} (n) = \underset{r = 1}{\sum^{n}} {c o s}^{- 1} (r f (x)) {∣ f (x) ∣⩽ 1 / n}

(4) S_{4} (n) = \underset{r = 1}{\sum^{n}} {s i n h}^{- 1} (r f (x))

(5) S_{5} (n) = \underset{r = 1}{\sum^{n}} {c o s h}^{- 1} (r f (x)) {f (x) ⩾ 1 / n}

(6) S_{6} (n) = \underset{r = 1}{\sum^{n}} {t a n h}^{- 1} (r f (x)) {∣ f (x) ∣< 1 / n}

Commented by prakash jain last updated on 05/Aug/15

Since most of these sums can be written

as addition . The sum is simply product .

Am i missing something here ?

Commented by 112358 last updated on 03/Aug/15

L e t y = {s i n h}^{- 1} x

\Rightarrow s i n h y = x

∵ s i n h y = \frac{e^{y} - e^{- y}}{2} \Rightarrow \frac{e^{y} - e^{- y}}{2} = x

e^{y} - e^{- y} = 2 x

\times e^{y} :\Rightarrow e^{2 y} - 2 {x e}^{y} - 1 = 0

∴ e^{y} = \frac{2 x \pm \sqrt{4 x^{2} + 4}}{2} = x \pm \sqrt{x^{2} + 1}

\Rightarrow y = l n (x \pm \sqrt{x^{2} + 1})

\sqrt{x^{2} + 1} > \sqrt{x^{2}} = x

x - \sqrt{x^{2} + 1} < 0 \Rightarrow y \in C \Leftrightarrow y = l n (x - \sqrt{x^{2} + 1})

∴ f o r y \in R, y = l n (x + \sqrt{x^{2} + 1}) [∵ \sqrt{x^{2} + 1} > - x]

i . e {s i n h}^{- 1} x = l n (x + \sqrt{x^{2} + 1})

A l s o, {t a n h}^{- 1} x = \frac{1}{2} l n (\frac{1 + x}{1 - x}) ∣ x ∣< 1

{c o s h}^{- 1} x = l n (x + \sqrt{x^{2} - 1}) x ⩾ 1

Commented by 112358 last updated on 05/Aug/15

I^{'} m w o n d e r i n g w h i c h o f t h e s e

s u m s h a s a n e x p l i c i t f o r m u l a

f o r i t s e v a l u a t i o n a n d t h e w a y s

b y w h i c h t h e y c a n b e f o u n d . T h i s

p o s t i s p u r e l y c o n s e q u e n t o f m y

c u r i o s i t y o f w h i c h s e r i e s o f

i n v e r s e t r i g o n o m e t r i c f u n c t i o n s

a n d i n v e r s e h y p e r b o l i c f u n c t i o n s

h a v e f o r m u l a e i n f i n d i n g t h e i r

s u m s . I {h a v e n}^{'} t y e t e x e c u t e d a

f u l l s c a l e i n v e s t i g a t i o n o f t h e s e

s u m s . I^{'} v e o n l y t r i e d o b t a i n i n g a n

a n s w e r f o r S_{1} (n) f o r n > 2 b u t t o

n o a v a i l . W h a t t o p i c (s) c a n I

r e a d u p o n t o h e l p m e w i t h t h i s

q u e s t i o n ?

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