Question

`\begin{gathered} \rm Prove \: that :- \\ { { \rm { f ( x ) = f ( 0 ) + (x)/(1!) f'(0) + (x²)/(2!) f''(0) + (x³)/(3!) f'''(0) + \cdot \cdot \cdot \cdot \cdot \cdot \cdot }}}\end{gathered}`

​

Answer 1

The series you had provided is generally the Maclaurin's series , and we want to prove the series , this series is very very much useful in calculus , especially when you have to substitute a function by it's series . So now let's start !

Consider the function f(x) as series of a polynomials of nth (n ≥ 1) degree as

`{\boxed{\bf{f(x)=a_(0)+a_(1)x+a_(2)x^(2)+a_(3)x^(3)+a_(4)x^(4)+\cdots \cdots \infty}}}`

Now , at x = 0

`{:\implies \quad \sf f(0)=a_(0)+a_(1)(0)+a_(2)(0)^(2)+a_(3)(0)^(3)+a_(4)(0)^(4)+\cdots \cdots \infty}`

`{\boxed{:\implies \quad \sf f(0)=a_(0)}}`

Now , again consider the function that we assumed

`{:\implies \quad \sf f(x)=a_(0)+a_(1)x+a_(2)x^(2)+a_(3)x^(3)+a_(4)x^(4)+\cdots \cdots \infty}`

Differentiating both sides w.r.t.x will yield

`{:\implies \quad \sf f^(\prime)(x)=0+a_(1)* 1+a_(2)* 2* x+a_(3)* 3* x^(2)+a_(4)* 4* x^(3)+\cdots \cdots \infty}`

`{:\implies \quad \sf f^(\prime)(x)=a_(1)+2a_(2)x+3a_(3)x^(2)+4a_(4)x^(3)\cdots \cdots \infty}`

Now , put x = 0 ;

`{:\implies \quad \sf f^(\prime)(0)=a_(1)+2a_(2)(0)+3a_(3)(0)^(2)+4a_(4)(0)^(3)\cdots \cdots \infty}`

`{:\implies \quad \sf f^(\prime)(0)=a_(1)}`

`{\boxed{:\implies \quad \sf a_(1)=(f^(\prime)(0))/(1!)}}`

Now , consider ;

`{:\implies \quad \sf f^(\prime)(x)=a_(1)+2a_(2)x+3a_(3)x^(2)+4a_(4)x^(3)\cdots \cdots \infty}`

Differentiating both sides w.r.t.x now ;

`{:\implies \quad \sf f^(\prime \prime)(x)=0+2* a_(2)* 1+3* a_(3)* 2* x+4* a_(4)* 3* x^(2)\cdots \cdots \infty}`

`{:\implies \quad \sf f^(\prime \prime)(x)=2a_(2)+6a_(3)x+12a_(4)x^(2)+\cdots \cdots \infty}`

Now , putting x = 0 will yield ;

`{:\implies \quad \sf f^(\prime \prime)(0)=2a_(2)+6a_(3)(0)+12a_(4)(0)^(2)+\cdots \cdots \infty}`

`{:\implies \quad \sf f^(\prime \prime)(0)=2a_(2)}`

`{\boxed{:\implies \quad \sf a_(2)=(f^(\prime \prime)(0))/(2!)}}`

Now , similarly we can prove that ;

`{\boxed{:\implies \quad \sf a_(n)=(f^(n)(0))/(n!)}}`

Now , as we had considered ;

`{:\implies \quad \sf f(x)=a_(0)+a_(1)x+a_(2)x^(2)+a_(3)x^(3)+a_(4)x^(4)+\cdots \cdots \infty}`

Now , putting the values we obtained above ;

`{:\implies \quad \sf f(x)=f(0)+(x)/(1!)f^(\prime)(0)+(x^2)/(2!)f^(\prime \prime)(0)+(x^3)/(3!)f^(\prime \prime \prime)(0)+(x^4)/(4!)f^(\prime \prime \prime \prime)(0)+\cdots \cdots (x^n)/(n!)f^(n)(0)+\cdots \cdots \infty}`

Hence , Proved