To find the Maclaurin series expansion of e^x, we can use the formula:

e^x = 1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + ...

Step 1: Write down the general form of the Maclaurin series expansion for e^x.

e^x = a0 + a1x + a2x^2 + a3x^3 + a4x^4 + ...

Step 2: Determine the coefficients a0, a1, a2, a3, a4, and so on.

To find the coefficients, we can use the formula:

an = f^n(0)/n!

where f^n(0) represents the nth derivative of e^x evaluated at x = 0.

Step 3: Calculate the derivatives of e^x.

The derivatives of e^x are:

f^0(x) = e^x
f^1(x) = e^x
f^2(x) = e^x
f^3(x) = e^x
f^4(x) = e^x
...

Step 4: Evaluate the derivatives at x = 0.

When x = 0, all the derivatives of e^x are equal to 1.

f^0(0) = 1
f^1(0) = 1
f^2(0) = 1
f^3(0) = 1
f^4(0) = 1
...

Step 5: Substitute the values into the formula for the coefficients.

Using the formula an = f^n(0)/n!, we have:

a0 = f^0(0)/0! = 1/1 = 1
a1 = f^1(0)/1! = 1/1 = 1
a2 = f^2(0)/2! = 1/2 = 1/2
a3 = f^3(0)/3! = 1/6 = 1/6
a4 = f^4(0)/4! = 1/24 = 1/24
...

Step 6: Write down the Maclaurin series expansion of e^x.

Using the coefficients we found, the Maclaurin series expansion of e^x is:

e^x = 1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + ...

This expansion can be used to approximate the value of e^x for any value of x.

Question

To find the Maclaurin series expansion of e^x, we can use the formula:

e^x = 1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + ...

Step 1: Write down the general form of the Maclaurin series expansion for e^x.

e^x = a0 + a1x + a2x^2 + a3x^3 + a4x^4 + ...

Step 2: Determine the coefficients a0, a1, a2, a3, a4, and so on.

To find the coefficients, we can use the formula:

an = f^n(0)/n!

where f^n(0) represents the nth derivative of e^x evaluated at x = 0.

Step 3: Calculate the derivatives of e^x.

The derivatives of e^x are:

f^0(x) = e^x
f^1(x) = e^x
f^2(x) = e^x
f^3(x) = e^x
f^4(x) = e^x
...

Step 4: Evaluate the derivatives at x = 0.

When x = 0, all the derivatives of e^x are equal to 1.

f^0(0) = 1
f^1(0) = 1
f^2(0) = 1
f^3(0) = 1
f^4(0) = 1
...

Step 5: Substitute the values into the formula for the coefficients.

Using the formula an = f^n(0)/n!, we have:

a0 = f^0(0)/0! = 1/1 = 1
a1 = f^1(0)/1! = 1/1 = 1
a2 = f^2(0)/2! = 1/2 = 1/2
a3 = f^3(0)/3! = 1/6 = 1/6
a4 = f^4(0)/4! = 1/24 = 1/24
...

Step 6: Write down the Maclaurin series expansion of e^x.

Using the coefficients we found, the Maclaurin series expansion of e^x is:

e^x = 1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + ...

This expansion can be used to approximate the value of e^x for any value of x.

Knowee AI · Accepted Answer

To find the Maclaurin series expansion of e^x, we can use the formula:

e^x = 1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + ...

Step 1: Write down the general form of the Maclaurin series expansion for e^x.

e^x = a0 + a1x + a2x^2 + a3x^3 + a4x^4 + ...

Step 2: Determine the coefficients a0, a1, a2, a3, a4, and so on.

To find the coefficients, we can use the formula:

an = f^n(0)/n!

where f^n(0) represents the nth derivative of e^x evaluated at x = 0.

Step 3: Calculate the derivatives of e^x.

The derivatives of e^x are:

f^0(x) = e^x
f^1(x) = e^x
f^2(x) = e^x
f^3(x) = e^x
f^4(x) = e^x
...

Step 4: Evaluate the derivatives at x = 0.

When x = 0, all the derivatives of e^x are equal to 1.

f^0(0) = 1
f^1(0) = 1
f^2(0) = 1
f^3(0) = 1
f^4(0) = 1
...

Step 5: Substitute the values into the formula for the coefficients.

Using the formula an = f^n(0)/n!, we have:

a0 = f^0(0)/0! = 1/1 = 1
a1 = f^1(0)/1! = 1/1 = 1
a2 = f^2(0)/2! = 1/2 = 1/2
a3 = f^3(0)/3! = 1/6 = 1/6
a4 = f^4(0)/4! = 1/24 = 1/24
...

Step 6: Write down the Maclaurin series expansion of e^x.

Using the coefficients we found, the Maclaurin series expansion of e^x is:

e^x = 1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + ...

This expansion can be used to approximate the value of e^x for any value of x.

Maclaurin’s series expansion of e^x