simplify-A-n-k-0-n-k-2-C-n-k-cos-k-and-B-n-k-0-n-k-2-C-n-k-sin-k-

Question Number 54675 by maxmathsup by imad last updated on 09/Feb/19

s i m p l i f y A_{n} = \sum_{k = 0}^{n} k^{2} C_{n}^{k} c o s (k θ) a n d B_{n} = \sum_{k = 0}^{n} k^{2} C_{n}^{k} s i n (k θ) .

Commented by maxmathsup by imad last updated on 10/Feb/19

w e h a v e A_{n} + {i B}_{n} = \sum_{k = 0}^{n} k^{2} C_{n}^{k} {(e^{i θ})}^{k} b u t w e h a v e p r o v e d t h a t

\sum_{k = 0}^{n} k^{2} C_{n}^{k} z^{k} = n z {(z + 1)}^{n - 1} + n (n - 1) z^{2} {(z + 1)}^{n - 2} \Rightarrow

\sum_{k = 0}^{n} k^{2} C_{n}^{k} {(e^{i θ})}^{k} = n e^{i θ} {(1 + e^{i θ})}^{n - 1} + n (n - 1) e^{2 i θ} {(e^{i θ} + 1)}^{n - 2} b u t

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

= 2 c o s (\frac{θ}{2}) e^{i \frac{θ}{2}} \Rightarrow A_{n} + {i B}_{n} = n e^{i θ} 2^{n - 1} {c o s}^{n - 1} (\frac{θ}{2}) e^{i \frac{n θ}{2}}

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

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

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

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

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

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

A_{n} = (n 2^{n - 1} {c o s}^{n - 1} (\frac{θ}{2}) + n (n - 1) 2^{n - 2}) c o s (\frac{(n + 2) θ}{2}) a n d

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

Answered by Smail last updated on 09/Feb/19

A_{n} + {i B}_{n} = S^{″} (θ)

S^{″} (θ) = \underset{k = 0}{\sum^{n}} k^{2} C_{n}^{k} e^{i θ k}

S^{'} (θ) = \frac{1}{i} \underset{k = 0}{\sum^{n}} {k C}_{n}^{k} e^{i θ k} + c_{1}

S (θ) = - \underset{k = 0}{\sum^{n}} C_{n}^{k} e^{i θ k} + θ c_{1} + c_{2}

S (θ) = - {(1 + e^{i θ})}^{n} + θ c_{1} + c_{2}

S^{'} (θ) = - {i n e}^{i θ} {(1 + e^{i θ})}^{n - 1} + c_{1}

S^{″} (θ) = {n e}^{i θ} {(1 + e^{i θ})}^{n - 1} + n (n - 1) e^{2 i θ} {(1 + e^{i θ})}^{n - 2}

= {n e}^{i θ} {(1 + e^{i θ})}^{n - 2} (1 + e^{i θ} + (n - 1) e^{i θ})

A_{n} + {i B}_{n} = {n e}^{i θ} {(1 + e^{i θ})}^{n - 2} (1 + {n e}^{i θ})

= {n e}^{i θ} {(1 + c o s θ + i s i n θ)}^{n - 2} (1 + {n e}^{i θ})

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

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

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

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

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

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

A_{n} = 2^{n - 2} {n c o s}^{n - 2} \frac{θ}{2} (c o s \frac{n θ}{2} + n c o s \frac{n + 2}{2} θ)

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

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