Illustration 5

Prove that C₀² + C₁² + C₂²+..............C_n² =

Ans : We know .

(1 + x)ⁿ = C₀+ C₁ C₂ x².......+C_n-1x^n-1 + C_nxⁿ

Also. (x + 1)ⁿ = C₀ xⁿ + C₁ x^n-1 + C₂ x^n-2+........... + C_n-1x + C_n

Now Let us multiply the two expansions and compare the coefficient of xⁿ on both side .

(1 + x)ⁿ (x - 1)ⁿ = (C₀ + C₁ x + C₂ x² ........+C_nx + xⁿ) X

(C₀ xⁿ + C₁ x^n-1 + C₂ x^n-2+........... + C_n-1x + C_n )

Coefficient of xⁿ in (1 + n)ⁿ (1 + n)ⁿ

=> Coeff. of xⁿ in (1 + x)²ⁿ

          = 2n_Cn

Coeff. of xⁿ in ( C₀ + C₁ x+ C₂ x² +......C_n xⁿ ) X

(C₀xⁿ + C₁xⁿ + ..........+C_n)

= C₀² + C₁ ² + C₂²+............+ C_n-1² + C_n²

C₀² + C₁ ² + C₂²+............+ C_n-1² + C_n²

So, Comparing the coefficients un the 2 sidse we get .

C₀² + C₁² + C₂²+............+ C_n-1² + C_n²= 2n _Cn

=

(2) Sum of sevies by use of differentiation:

When numericals occur as product of bionomial Coefficients this method is used .

Illustration 6

Find thesum of the following sevies.

C₀ + 2C₁ + 3C₂ +...............+ (n + 1) C_n?

Ans We know that

(1 + x)ⁿ = C₀ + C₁x +C₂ x² + ..................+C_n xⁿ

So, Multiplying both sides with x we get

x(1 + x)ⁿ = C₀ x + C₁ x² + C₂ x³ + ........+C_n x ⁿ⁺¹

Now differentiating both sides with respect x we get

xn(1 + x)^n-1 + (1 + x)ⁿ = C₀ + 2C₁x + 3c₂ x²+...........+(n + 1) C_n xⁿ

Now put x = 1

So, n 2^n-1 + 2ⁿ = C₀ + 2c₁ + 3C₂+...........+ (n + 1)C_n

So, C₀ + 2c₁ + 3C₂+...........+(n + 1)C_n = (n + 2)2^n-1

Dumb Question:- Why did we multiplied bionomial expansion with x imitially ?

Ans Observe taht the numreicals with bionomial coefficints were one more in value that the corresponding power of x, So we needed to multiply the expassion by x.

(3) Sum of series by use of integration

Whan nummericals occuv as the denominator of the bionomial Coefficient we apply this method .

Illustration 7.

Find the sum of series 2C_o + 2 ²

And

We know

(1 + x) ⁿ = C₀ + C₁ x + C₂ x² +..........+ C_nxⁿ

Integrating both sides with respect to x we get

(C₀ + C₁ x + C₂ x² +..........+ C_nx^{n) dx}

=>

2C_o + 2²

=

(4) Sum of the series by equation real and imaginary parts.

Somctimes the use of complex nos make the calculation very easy .