Evaluate-1-2-1-3-1-4-1-

Question Number 17991 by alex041103 last updated on 13/Jul/17

E v a l u a t e \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + 4 \sqrt{1 + \dots}}}}

Commented by alex041103 last updated on 13/Jul/17

j u s t f o r f u n

w h o e v e r w a n t s c a n t r y t o g e n e r a l i z e

\sqrt{A + B_{1} \sqrt{A + B_{2} \sqrt{A + B_{3} \sqrt{A + \dots}}}}

Commented by alex041103 last updated on 13/Jul/17

A n s w e r s t o Q . 17989 a n d Q . 17991 o n

15^{t h} J u l y

Commented by prakash jain last updated on 13/Jul/17

3 = \sqrt{1 + 8}

= \sqrt{1 + 2 \cdot 4}

= \sqrt{1 + 2 \sqrt{16}}

= \sqrt{1 + 2 \sqrt{1 + 15}}

= \sqrt{1 + 2 \sqrt{1 + 3.5}}

= \sqrt{1 + 2 \sqrt{1 + 3 . \sqrt{1 + 24}}}

= \sqrt{1 + 2 \sqrt{1 + 3 . \sqrt{1 + 4.6}}}

= \sqrt{1 + 2 \sqrt{1 + 3 . \sqrt{1 + 4 . \sqrt{1 + 5 . \sqrt{1 + 6 \sqrt{1 +} . .}}}}}

Answered by prakash jain last updated on 13/Jul/17

3

Commented by alex041103 last updated on 13/Jul/17

B o n u s : T h i s i n f i n i t e b e a u t y w a s d i s c o v e r e d

b y t h e g e n i u s i n d i a n m a t h m a t i t i o n

S r i n i v a s a R a m a n u d j a n

Answered by alex041103 last updated on 14/Jul/17

G e n e r a l i z a t i o n

l e t A = x^{2} a n d B_{n + 1} = x + B_{n}

B_{1} = \sqrt{B_{1}^{2}} = \sqrt{A + (B_{1}^{2} - A)} =

= \sqrt{A + (B_{1} - x) (B_{1} + x)} =

= \sqrt{A + B_{0} \sqrt{A + B_{2}^{2} - x^{2}}} =

= \sqrt{A + B_{0} \sqrt{A + (B_{2} - x) (B_{2} + x)}} =

= \sqrt{A + B_{0} \sqrt{A + B_{1} \sqrt{B_{3}^{2}}}} =

= \dots . =

= \sqrt{A + B_{0} \sqrt{A + B_{1} \sqrt{A + B_{2} \sqrt{A + B_{3} \sqrt{\dots}}}}}

B_{1} = \sqrt{A + B_{0} \sqrt{A + B_{1} \sqrt{A + B_{2} \sqrt{A + B_{3} \sqrt{\dots}}}}}

E x a m p l e :

A = 1 a n d B_{1} = 4

4 = \sqrt{1 + 3 \sqrt{1 + 4 \sqrt{1 + 5 \sqrt{1 + 6 \sqrt{\dots}}}}}