Find-x-1-1-x-sin-1-x-

Question Number 58827 by jimful last updated on 30/Apr/19

Find \sum_{x = 1}^{\infty} (\frac{1}{x} - \sin (\frac{1}{x}))

Commented by tanmay last updated on 30/Apr/19

t a k i n g t h e h e l p o f g r a p h w e c a n f i n d t h e v a l u e

Commented by jimful last updated on 01/May/19

thanks . I need to learn more

Commented by maxmathsup by imad last updated on 30/Apr/19

w e h a v e s i n x = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{2 n + 1}}{(2 n + 1)!} = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \dots \Rightarrow

x - \frac{x^{3}}{6} ⩽ s i n x ⩽ x \Rightarrow \frac{1}{x} - \frac{1}{6 x^{3}} ⩽ s i n (\frac{1}{x}) ⩽ \frac{1}{x}

\Rightarrow - \frac{1}{x} ⩽ - s i n (\frac{1}{x}) ⩽ - \frac{1}{x} + \frac{1}{6 x^{3}} \Rightarrow 0 ⩽ \frac{1}{x} - s i n (\frac{1}{x}) ⩽ \frac{1}{6 x^{3}} \Rightarrow

0 ⩽ \sum_{x = 1}^{\infty} (\frac{1}{x} - s i n (\frac{1}{x})) ⩽ \frac{1}{6} \sum_{x = 1}^{\infty} \frac{1}{x^{3}} \Rightarrow 0 ⩽ S ⩽ \frac{1}{6} ξ (3) \dots

Answered by tanmay last updated on 30/Apr/19

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

g (x) = s i n (\frac{1}{x}) g (1) = s i n (1) = 0.841

f (1) - g (1) = 1 - 0.841 = 0.159

f (2) = \frac{1}{2} = 0.5

g (2) = s i n (\frac{1}{2}) = 0.479

f (2) - g (2) = 0.5 - 0.479 = 0.021

△ = f (x) - g (x)

t h u s w e s e e a s w e i n c r e a s e t h e v a l u e o f x

△ \to 0

s o \underset{x = 1}{\sum^{\infty}} \frac{1}{x} - s i n (\frac{1}{x}) = 0.159 + 0.021 + ϵ

\underset{x = 1}{\sum^{\infty}} \frac{1}{x} - s i n (\frac{1}{x}) = 0.18 + ϵ

ϵ = s m a l l p o s i t i v e n u m b e r

i h a v e t r i e d t o f i n d . . i t i s n o t e x a c t s o l u t i o n

b u t i n t e r p r e t a t i o n o f t h e p r o b l e m \dots

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