0-lt-x-lt-1-1-1-x-1-2x-1-x-2-4x-3-1-x-4-8x-7-1-x-8-16x-15-1-x-16-evaluate-the-previous-summation-

Question Number 195651 by York12 last updated on 06/Aug/23

0 < x < 1

\frac{1}{1 + x^{1}} + \frac{2 x}{1 + x^{2}} + \frac{4 x^{3}}{1 + x^{4}} + \frac{8 x^{7}}{1 + x^{8}} + \frac{16 x^{15}}{1 + x^{16}} + \dots . + \infty

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

Answered by witcher3 last updated on 06/Aug/23

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

\int_{0}^{t} f (x) dx = \sum_{n ⩾ 0} \ln (1 + t^{2^{n}}) = \ln (\prod_{n ⩾ 0} (1 + t^{2^{n}}))

1 + x = \frac{1 - x^{2}}{1 - x}

= \ln (\prod_{n ⩾ 0} \frac{1 - t^{2^{n + 1}}}{1 - t^{2^{n}}}) = \underset{N \to \infty}{\lim l n} (\underset{0}{\prod^{N}} \frac{1 - t^{2^{n + 1}}}{1 - t^{2^{n}}})

= \underset{N \to \infty}{\lim l n} (\frac{1 - t^{2^{N + 1}}}{1 - t}) = - \ln (1 - t) = \int_{0}^{t} f (x) dx

f (x) = \frac{1}{1 - x}

Commented by York12 last updated on 07/Aug/23

s h o u l d I g e t t h r o u g h c a l c u l u s

o r I c a n s k i p i t a n d s t u d y r e a l a n a l y s i s

Commented by witcher3 last updated on 07/Aug/23

in Maths You {can}^{'} t skip lesson all are usufull

Geometrie related algebra withe analysis

withe number theory

geometrie & algebra Geometric - algebra

number theorie + analysis = modular forms

algebra and analysis = K Theory

k theory is not used so much one if the hardest Topic

in Maths

Commented by York12 last updated on 08/Aug/23

t h a n k s s o m u c h s i r

Commented by witcher3 last updated on 09/Aug/23

withe Pleasur God bless You

Commented by York12 last updated on 10/Aug/23

Facebook Tweet Pin