Introduction to differentiation of Power series:
Let us see the introduction of Differentiation of power series. The differentiation of power series occurs primarily in analysis, but also occurs in combinatorics. The real numbers can also be viewed as the familiar decimal notation. Example for differentiation of power series with the argument x fixed at 1⁄10. We discuss the formulas for differentiation of power series. I like to share this Differentiation Math with you all through my article.
Definition of Differentiation of Power Series
The differentiation of power series convergence of a positive radius and it is analytic on the interior of its convergence of region.Let us see the example problems for Differentiation of power series. Please express your views of this topic Derivative Formulas by commenting on blog.
Example Problems for Differentiation of Power Series:
Ex:1 Find ex with an error less than 1 / 4!
Sol:
Step 1:
For all x we have ex= 1= x+ +x2/2! +x3/3! +x4/4! +……..
Step 2:
Substitute x=-1 then,
Step 3:
e-1= 1-1+1/2!-1/3! +1/4! +….
First omitted term because this is an alternating decreasing series so the error is smaller.
Step 4:
Answer is:
e-1= 1-1+1/2!-1/3!
Ex:2 Solve the differentiation of power series for the following polynomial function f(x) = 2x2 + 4x + 3.
Sol:
Step 1:
The given function is f(x) = 2x2 + 4x + 3.
Step 2:
The above equation can be written as a power series around the center c = 0 as the,
f’(x) =3+4x+4x2+0x3+…….
Around the center c = 1
f’(x)=6+8(x-1)+4(x-1)2+0(x-1)3+………….
Step 3:
Answer is:
f’(x)=6+8(x-1)+4(x-1)2+0(x-1)3+………….
Ex:3 Solve the differentiation of power series for the following polynomial function f(x) = 4x2 + 8x + 6.
Sol:
Step 1:
The given function is f(x) = 4x2 + 8x + 6.
Step 2:
The above equation can be written as a power series around the center c = 0 as the,
f’(x) =6+8x+8x2+0x3+…….
Around the center c = 1
f’(x)=12+16(x-1)+8(x-1)2+0(x-1)3+………….
Step 3:
Answer is:
f’(x)=12+16(x-1)+8(x-1)2+0(x-1)3+……
Let us see the introduction of Differentiation of power series. The differentiation of power series occurs primarily in analysis, but also occurs in combinatorics. The real numbers can also be viewed as the familiar decimal notation. Example for differentiation of power series with the argument x fixed at 1⁄10. We discuss the formulas for differentiation of power series. I like to share this Differentiation Math with you all through my article.
Definition of Differentiation of Power Series
The differentiation of power series convergence of a positive radius and it is analytic on the interior of its convergence of region.Let us see the example problems for Differentiation of power series. Please express your views of this topic Derivative Formulas by commenting on blog.
Example Problems for Differentiation of Power Series:
Ex:1 Find ex with an error less than 1 / 4!
Sol:
Step 1:
For all x we have ex= 1= x+ +x2/2! +x3/3! +x4/4! +……..
Step 2:
Substitute x=-1 then,
Step 3:
e-1= 1-1+1/2!-1/3! +1/4! +….
First omitted term because this is an alternating decreasing series so the error is smaller.
Step 4:
Answer is:
e-1= 1-1+1/2!-1/3!
Ex:2 Solve the differentiation of power series for the following polynomial function f(x) = 2x2 + 4x + 3.
Sol:
Step 1:
The given function is f(x) = 2x2 + 4x + 3.
Step 2:
The above equation can be written as a power series around the center c = 0 as the,
f’(x) =3+4x+4x2+0x3+…….
Around the center c = 1
f’(x)=6+8(x-1)+4(x-1)2+0(x-1)3+………….
Step 3:
Answer is:
f’(x)=6+8(x-1)+4(x-1)2+0(x-1)3+………….
Ex:3 Solve the differentiation of power series for the following polynomial function f(x) = 4x2 + 8x + 6.
Sol:
Step 1:
The given function is f(x) = 4x2 + 8x + 6.
Step 2:
The above equation can be written as a power series around the center c = 0 as the,
f’(x) =6+8x+8x2+0x3+…….
Around the center c = 1
f’(x)=12+16(x-1)+8(x-1)2+0(x-1)3+………….
Step 3:
Answer is:
f’(x)=12+16(x-1)+8(x-1)2+0(x-1)3+……
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