Study-f-x-n-1-x-n-sin-nx-n-Find-out-n-1-1-n-sin-n-n-and-n-1-sin-n-n-

Question Number 79126 by ~blr237~ last updated on 22/Jan/20

S t u d y f (x) = \underset{n = 1}{\sum^{\infty}} \frac{x^{n} s i n (n x)}{n}

F i n d o u t \underset{n = 1}{\sum^{\infty}} {(- 1)}^{n} \frac{s i n (n)}{n} a n d \underset{n = 1}{\sum^{\infty}} \frac{s i n (n)}{n}

Commented by mathmax by abdo last updated on 25/Jan/20

l e t s (x) = \sum_{n = 1}^{\infty} \frac{s i n (n x)}{n} w e h a v e s (x) = I m (\sum_{n = 1}^{\infty} \frac{e^{i n x}}{n})

l e t w (x) = \sum_{n = 1}^{\infty} \frac{e^{i n x}}{n} \Rightarrow w^{^{'}} (x) = \sum_{n = 1}^{\infty} \frac{i n e^{i n x}}{n}

= i \sum_{n = 1}^{\infty} {(e^{i x})}^{n} = i {\sum_{n = 0}^{\infty} {(e^{i x})}^{n} - 1}

= i {\frac{1}{1 - e^{i x}} - 1} = i {\frac{1}{1 - c o s x - i s i n x} - 1}

= i {\frac{1 - c o s x + i s i n x}{{(1 - c o s x)}^{2} + {s i n}^{2} x} - 1} = i (\frac{1 - c o s x}{{(1 - c o s x)}^{2} + {s i n}^{2} x}) - \frac{s i n x}{{(1 - c o s x)}^{2} + {s i n}^{2} x} - i

= i (\frac{1 - c o s x}{2 - 2 c o s x} - 1) - \frac{s i n x}{2 - 2 c o s x} = - \frac{i}{2} - \frac{s i n x}{4 {s i n}^{2} (\frac{x}{2})}

= - \frac{i}{2} - \frac{c o s (\frac{x}{2})}{2 s i n (\frac{x}{2})} \Rightarrow w (x) = - \frac{i x}{2} - \frac{1}{2} \int \frac{c o s (\frac{x}{2})}{s i n (\frac{x}{2})} d x + c (\frac{x}{2} = t)

w (x) = - \frac{i x}{2} - \int \frac{c o s t}{s i n t} d t = - i \frac{x}{2} - l n ∣ s i n (\frac{x}{2}) ∣ + c

w (π) = \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{n} = - l n (2) = - \frac{i π}{2} + c \Rightarrow c = - l n (2) + \frac{i π}{2} \Rightarrow

w (x) = - \frac{i x}{2} - l n ∣ s i n (\frac{x}{2}) ∣ - l n (2) + \frac{i π}{2} = - l n (2 s i n (\frac{x}{2}) ∣ + i \frac{π - x}{2}

I m (w (x) = \frac{π - x}{2} = s (x)

x = 1 \Rightarrow \sum_{n = 1}^{\infty} \frac{s i n (n)}{n} = \frac{π - 1}{2}

Commented by mathmax by abdo last updated on 25/Jan/20

f (x) = I m (\sum_{n = 1}^{\infty} \frac{e^{i n x} x^{n}}{n}) l e t

φ (x) = \sum_{n = 1}^{\infty} \frac{e^{i n x} x^{n}}{n} \Rightarrow φ (x) = \sum_{n = 1}^{\infty} \frac{{({x e}^{i x})}^{n}}{n} = \sum_{n = 1}^{\infty} \frac{z^{n}}{n}

w i t h z = x e^{i x} f o r ∣ z ∣=∣ x ∣⩽ w r g e t

\frac{d}{d z} (Σ \frac{z^{n}}{n}) = \sum_{n = 1}^{\infty} z^{n - 1} = \sum_{n = 0}^{\infty} z^{n} = \frac{1}{1 - z} \Rightarrow \sum_{n = 1}^{\infty} \frac{z^{n}}{n} = - l n (1 - z)

- l n (1 - {x e}^{i x}) = - l n (1 - x (c o s x + i s i n x))

= - l n (1 - x c o s x - i x s i n x) w e h s v e

∣ 1 - x c o s x - i x s i n x ∣ = \sqrt{{(1 - x c o s x)}^{2} + x^{2} {s i n}^{2} x}

= \sqrt{1 - 2 x c o s x + x^{2}} \Rightarrow 1 - x c o s x - i x s i n x

= \sqrt{x^{2} - 2 x c o s x + 1} e^{i s r c t a n (\frac{- x s i n x}{1 - x c o s x})} \Rightarrow

- l n (\dots) = - \frac{1}{2} l n (x^{2} - 2 x c o s x + 1) + i a r c t a n (\frac{x s i n x}{1 - x c o s x})

f (x) = I m (φ (x)) = a r c t a n (\frac{x s i n x}{1 - x c o s x})

x = - 1 \Rightarrow \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{n} s i n (n) = a r c t a n (\frac{s i n (1)}{1 + c o s (1)})

= a r c t a n (\frac{2 s i n (\frac{1}{2}) c o s (\frac{1}{2})}{2 {c o s}^{2} (\frac{1}{2})}) = \frac{1}{2}

Answered by mind is power last updated on 23/Jan/20

i f ∣ x ∣> 1

\lim_{n \to \infty} \frac{x^{n} s i n (n x)}{n}, {d i d}^{'} n t e x i s t e

\Rightarrow x \in [- 1, 1]

x = 1 w e u s e a b e l l e m m a

s i n c e \frac{1}{n} d e c r e a s e i n z e r o

\sum_{n ⩾ 1} s i n (n) = I m \sum_{N ⩾ n ⩾ 1} e^{i n} = {I m e}^{i} (\frac{1 - e^{i (N + 1)}}{1 - e^{i}})

= I m (e^{\frac{i}{2}} (\frac{e^{i \frac{N + 1}{2}} s i n (\frac{N + 1}{2})}{s i n (\frac{1}{2})}))

= \frac{s i n (\frac{N + 2}{2}) s i n (\frac{N + 1}{2})}{s i n (\frac{1}{2})} \Rightarrow∣ Σ s i n (n) ∣⩽ \frac{1}{∣ s i n (\frac{1}{2}) ∣}

b y a b e l \sum_{n ⩾ 1} \frac{s i n (n)}{n} c v

f o r - 1, \sum_{n ⩾ 1} \frac{{(- 1)}^{n} s i n (n)}{n}

s a m ∣ Σ {(- 1)}^{n} s i n (n) ∣< \infty, \frac{1}{n} d e c r e a s e \to 0 c v

f o r x \in [- a, a], 0 ⩽ a < 1

f (x) ⩽ \sum_{n ⩾ 1} \frac{∣ x ∣^{n}}{n} = - l n (1 - ∣ x ∣) e v i d e n t

f i s w e l l d e f i n d i n [- 1, 1]

f (x) = Σ \frac{x^{n} s i n (n)}{n} = Σ \frac{x^{n}}{n} I m (e^{i n})

= I m Σ \frac{x^{n} e^{i n}}{n} = I m Σ \frac{{({x e}^{i})}^{n}}{n} = I m {- l n (1 - {x e}^{i})}

= I m {- l n (1 - x c o s (1) - i x s i n (1))

1 - x c o s (1) - i x s i n (1) = \sqrt{1 + x^{2} - 2 x c o s (1)} e^{i (a r c t a n (\frac{- x s i n (1)}{1 - x c o s (1)})}

l n (1 - {x e}^{i}) = l n (\sqrt{1 + x^{2} - 2 x c o s (1)}) + i (a r c t a n (\frac{- x s i n (1)}{1 - x c o s (1)}))

I m {- l n (1 - {x e}^{i})} = - a r c t a n (\frac{- x s i n (1)}{1 - x c o s (1)}) = a r c t a n (\frac{x s i n (1)}{1 - x c o s (1)})

f (1) = \sum_{n ⩾ 1} \frac{s i n (n)}{n} = a r c t a n (\frac{s i n (1)}{1 - c o s (1)}) = a r c t a n (\frac{2 s i n (\frac{1}{2}) c o s (\frac{1}{2})}{2 {s i n}^{2} (\frac{1}{2})})

= a r c t a n (t a n (\frac{π}{2} - \frac{1}{2})) = \frac{π - 1}{2}

f (- 1) = \sum_{n ⩾ 1} \frac{{(- 1)}^{n} s i n (n)}{n} = a r c t a n (\frac{- s i n (1)}{1 + c o s (1)}) = - a r c t a n (t g (\frac{1}{2}))

= - \frac{1}{2}