Prove that 1n-12-n-2- 14-n-4-1n-2n-1-

L.H.S [ 1 n! + 1 2!( n 2 ) ! + 1 4!( n 2 ) ! +...........................; 1 n! [ ^ n C_ 0 +^ n C_ 2 +^ n C_ 4 +^ n C_ 6 +................ ...

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nCr Definitions and Properties

Prove that ...

Question

Prove that $\frac{1}{n!} + \frac{1}{2! (n - 2)!} + \frac{1}{4! (n - 4)!} + = \frac{1}{n!} 2^{n - 1}$ .

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1 + \frac{2 n}{3} + \frac{2 n (2 n + 2)}{3.6} + \frac{2 n (2 n + 2) (2 n + 4)}{3.6.9} + . . . . .

= 2^{n} {1 + \frac{n}{3} + \frac{n (n + 1)}{3.6} + \frac{n (n + 1) (n + 2)}{3.6.9} + . . . .}

\frac{C_{0}}{1} - \frac{C_{1}}{4} + \frac{C_{2}}{9} - . . . + \frac{(- 1)^{n} C_{n}}{n^{2} + 2 n + 1}

= \frac{1}{n + 1} + \frac{1}{2 n + 2} + . . . + \frac{1}{n^{2} + 2 n + 1} .

n \geq 1

1^{2} + 2^{2} + 3^{2} + 4^{2} + \dots + n^{2} = \frac{n (n + 1) (2 n + 1)}{6}

n \geq 1,

1^{2} + 2^{2} + 3^{2} + 4^{2} + . . . . + n^{2} = \frac{n (n + 1) (2 n + 1)}{6}

1 + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + . . . . . \frac{1}{1 + 2 + 3 + . . . . n} = \frac{2 n}{n + 1}

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