find-S-n-k-0-n-k-2-C-n-k-cos-2kx-interms-of-n-

Question Number 56962 by turbo msup by abdo last updated on 27/Mar/19

f i n d S_{n} = \sum_{k = 0}^{n} k^{2} C_{n}^{k} c o s (2 k x)

i n t e r m s o f n .

Commented by maxmathsup by imad last updated on 29/Mar/19

w e h a v e S_{n} = R e (\sum_{k = 0}^{n} k^{2} C_{n}^{k} e^{i 2 k x}) = R e (W_{n})

l e t p (x) = \sum_{k = 0}^{n} C_{n}^{k} e^{i 2 k x} = \sum_{k = 0}^{n} C_{n}^{k} {(e^{i 2 x})}^{k} = {(1 + e^{i 2 x})}^{n}

b u t p^{^{'}} (x) = 2 i \sum_{k = 0}^{n} k C_{n}^{k} e^{i 2 k x} a n d p^{^{″}} (x) = - 4 \sum_{k = 0}^{n} k^{2} C_{n}^{k} e^{i 2 k x} \Rightarrow

\sum_{k = 0}^{n} k^{2} C_{n}^{k} e^{i 2 k x} = - \frac{1}{4} p^{(2)} (x) w e h a v e p (x) = {(e^{i 2 x} + 1)}^{n} \Rightarrow

p^{^{'}} (x) = 2 i n {(e^{i 2 x} + 1)}^{n - 1} a n d p^{(2)} (x) = - 4 n (n - 1) {(e^{i 2 x} + 1)}^{n - 2}

b u t {(e^{i 2 x} + 1)}^{n - 2} = {(c o s (2 x) + i s i n (2 x) + 1)}^{n - 2}

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

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

\sum_{k = 0}^{n} k^{2} C_{n}^{k} e^{i 2 k x} = - \frac{1}{4} (- 4 n (n - 1)) {(2 c o s x)}^{n - 2} {c o s (n - 2) x + i s i n (n - 2) x}

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

S_{n} = n (n - 1) 2^{n - 2} {c o s}^{n - 2} (x) c o s ((n - 2) x) .

Answered by Smail last updated on 27/Mar/19

S_{n} (x) = \underset{k = 0}{\sum^{n}} k^{2} C_{n}^{k} c o s (2 k x) = R e (\underset{k = 0}{\sum^{n}} k^{2}^{n} C_{k} e^{2 i k x})

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

z^{'} (x) = \frac{1}{2 i} \underset{k = 0}{\sum^{n}} k^{n} C_{k} e^{2 i k x} + c

z (x) = - \frac{1}{4} \underset{k = 0}{\sum^{n}}^{n} C_{k} e^{2 i k x} + c x + a

= \frac{- 1}{4} {(1 + e^{2 i x})}^{n} + c x + a

z^{'} (x) = \frac{1}{2 i} \times {n e}^{2 i x} {(1 + e^{2 i x})}^{n - 1} + c

z^{″} (x) = n (e^{2 i x} {(1 + e^{2 i x})}^{n - 1} + (n - 1) e^{4 i x} {(1 + e^{2 i x})}^{n - 2})

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

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

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

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

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

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

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

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

S_{n} = 2^{n - 2} {n c o s}^{n - 2} x (c o s n x + n c o s (n + 2) x)

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