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Question Number 33131 by prof Abdo imad last updated on 11/Apr/18

1) f i n d \sum_{n = 1}^{\infty} \frac{e^{i n x}}{n (n + 1)}

2) f i n d t h e v a l u e o f \sum_{n ⩾ 1} \frac{s i n (n x)}{n (n + 1)}

a n d \sum_{n ⩾ 1} \frac{c o s (n x)}{n (n + 1)} .

Commented by prof Abdo imad last updated on 12/Apr/18

l e t f i n d S (x) = \sum_{n = 1}^{\infty} \frac{x^{n}}{n (n + 1)} t h e r a d i u s o f

c o n v e r g r n c e i s 1 a n d f o r x = \bar{+} 1 t h e s e r i e i s

a l s o c o n v e r g e n t w e h a v e f o r ∣ x ∣< 1

S (x) = \sum_{n = 1}^{\infty} (\frac{1}{n} - \frac{1}{n + 1}) x^{n} = \sum_{n = 1}^{\infty} \frac{x^{n}}{n} - \sum_{n = 1}^{\infty} \frac{x^{n}}{n + 1}

b u t \sum_{n = 1}^{\infty} \frac{x^{n}}{n} = - l n (1 - x)

\sum_{n = 1}^{\infty} \frac{x^{n}}{n + 1} = \sum_{n = 2}^{\infty} \frac{x^{n - 1}}{n} = \frac{1}{x} \sum_{n = 2}^{\infty} \frac{x^{n}}{n}

= \frac{1}{x} (- l n (1 - x) - x) = - \frac{1}{x} l n (1 - x) - 1 \Rightarrow

S (x) = - l n (1 - x) + \frac{1}{x} l n (1 - x) + 1

= (- 1 + \frac{1}{x}) l n (1 - x) + 1 \Rightarrow

S (x) = \frac{1 - x}{x} l n (1 - x) + 1 l e t c h a n g e x p e r e^{i x}

w e g e t \sum_{n = 1}^{\infty} \frac{e^{i n x}}{n (n + 1)} = \frac{1 - e^{i x}}{e^{i x}} l n (1 - e^{i x}) + 1

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

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

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

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

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

= - 2 i e^{- i \frac{x}{2}}

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

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

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

= l n (- i) + l n (2 s i n (\frac{x}{2}) + l n (e^{i \frac{x}{2}})

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

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

I m (S (e^{i x})) = - 2 i e^{- i \frac{x}{2}} (l n (2 s i n (\frac{x}{2}) + i \frac{x - π}{2})

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

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

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

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

- 2 s i n (\frac{x}{2}) l n (2 s i n (\frac{x}{2}) + i (π - x) s i n (\frac{x}{2}) \Rightarrow

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

\sum_{n = 1}^{\infty} \frac{c o s (n x)}{n (n + 1)} = (x - π) c o s (\frac{x}{2}) - 2 s i n (\frac{x}{2}) l n (2 s i n (\frac{x}{2})) + 1 .