Sum of the series - k-0 n - r-0 n-k - n- k - n-k- r - is

The correct options are B 3 ^ n C ∑ _ r=0 ^ n ( 1) ^ r C _ r 4 ^ r D ∑ _ r=0 ^ n _^ n C _ r 2 ^ r ∑_k=0 ^k=n∑_r=0 ^n k ^nC_k ^n kC_r ...

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Existence of Limit

Sum of the se...

Question

Sum of the series
$\sum_{k = 0}^{n} \sum_{r = 0}^{n - k} (\begin{matrix} n k \end{matrix}) (\begin{matrix} n - k r \end{matrix})$ is

2^{n}

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3^{n}

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\sum_{r = 0}^{n} {(- 1)}^{r} C_{r} 4^{r}

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\sum_{r = 0}^{n} {^{n} C}_{r} 2^{r}

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t_{n} = \sum_{r = 0}^{n} \frac{1}{{({^{n} C}_{r})}^{k}}

S_{n} = \sum_{r = 0}^{n} \frac{r}{{({^{n} C}_{r})}^{k}}

k \in Z^{+}

{cos}^{- 1} (\frac{S_{n}}{n t_{n}})

a_{n} = n \sum r = 0 \frac{1}{{^{n} C}_{r}}

n \sum r = 0 \frac{r}{{^{n} C}_{r}}

det ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} n \sum k = 0 k & n \sum k = 0^{n} C_{k} k^{2} n \sum k = 0^{n} C_{k} k & n \sum k = 0^{n} C_{k} 3^{k} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = 0

n . Then n \sum k = 0 \frac{^{n} C_{k}}{k + 1}

\infty \sum n = 0 \frac{1}{n!} [n \sum k = 0 (k + 1) \int_{0}^{1} 2^{- (k + 1) x} d x]

S_{1} = \sum_{k = 0}^{n} {^{n} C}_{k} cos (k x) cos (n - k) x

S_{2} = \sum_{k = 0}^{n} {^{n} C}_{k} sin (k x) sin (n - k) x

S_{1} - S_{2}

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