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in Continuity and Differentiability by (36.2k points)
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If (1 + x)n = C0 + C1x + C2x2 +....+ Cnxn, then prove that

(i) C1 + 2C+ ...+ nC=  n.2n-1

(ii) C1 - 2C+ 3C3 - ...+(-1)n nCn=  0

1 Answer

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Best answer

We have,

(1 + x)n = C0 + C1x + C2x2 +....+ Cnxn,

Differentiating both sides with respect to x, we have

n(1 + x)n-1 = C1 + C2x + 3C3x2....+ nCnxn-1,

Putting x = 1 and x = -1 successively, we have

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