-n-1Cn-0-Cn-1- - -Cn-n-Pn-n- -ek-m-Find k-m -

We have nbsp; nbsp; nbsp; ∑^∞_n=1C(n,0)+C(n,1)+ ... +C(n,n)/P(n,n) nbsp; nbsp; nbsp; nbsp; nbsp; nbsp; nbsp; =∑^∞_n=1^nC_0 + ...

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Logarithmic Function

∑∞n=1Cn,0+Cn,...

Question

$\infty \sum n = 1 \frac{C (n, 0) + C (n, 1) + . . . + C (n, n)}{P (n, n)}$ = $= e^{k} + m$ .Find $k + m$ ?

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Solution

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^{n} C_{n} +^{n + 1} C_{n} +^{n + 2} C_{n} + . . . +^{n + k} C_{n + 1} =

C_{n} C_{n - 2} + C_{n - 1} C_{n - 3} + C_{n - 2} C_{n - 4} + . . . + C_{3} C_{1} + C_{2} C_{0} =^{2 n} C_{n - λ}

C_{i} =^{n} C_{i}, λ \in N

λ

\frac{1}{m!} C_{0} + \frac{n}{(m + 1)!} C_{1} + \frac{n (n - 1)}{(m + 2)!} C_{2} + . . . . . + \frac{n (n - 1) . . .2 .1}{(m + n)!} C_{n} = \frac{(m + n + 1) (m + n + 2) . . . (m + 2 n)}{(m + n)!}

r \geq 0

(\begin{matrix} m r \end{matrix}) C_{0} + (\begin{matrix} m r - 1 \end{matrix}) C_{1} + . . . . . . (\begin{matrix} m 0 \end{matrix}) C_{r} = (\begin{matrix} m + n r \end{matrix})

{^{n} C}_{0} {^{n} C}_{n} + {^{n} C}_{1} {^{n} C}_{n - 1} + . . . . + {^{n} C}_{n} {^{n} C}_{0}

f (x) = ⎡ ⎢ ⎣ \begin{matrix} n & n + 1 & n + 2 {^{n} P}_{n} & {^{n + 1} P}_{n + 1} & {^{n + 2} P}_{n + 2} {^{n} C}_{n} & {^{n + 1} C}_{n + 1} & {^{n + 2} C}_{n + 2} \end{matrix} ⎤ ⎥ ⎦

| f (x) |

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