
How do you find the Maclaurin series for ?
Answer
412.8k+ views
Hint: We need to find the Maclaurin series for . We know that Maclaurin series is given by
To calculate the Maclaurin series, we need to calculate the first, second, third up to derivative of the given equation. After finding the derivative we need to calculate the value of the derivatives at zero. Putting all the values in the Maclaurin series of , we will get the Maclaurin series for .
Complete step by step answer:
We have to find the Maclaurin series of .
Let .
As we know that the Maclaurin series is given by
Putting in , we get
On simplifying, we get
As we know, from the product rule of differentiation,
Differentiating with respect to using the product rule of differentiation, we get
On simplification, we get
Putting , we get
Putting the value of , and , we get
Now, differentiating with respect to , we get
On simplification, we get
On simplification, we get
Putting , we get
Now, differentiating with respect to , we get
On simplification, we get
Putting , we get
From and , we get
On simplification, we get
Therefore, the Maclaurin series for is .
Note:
To find the Taylor series and Maclaurin series, the function must be infinitely times differentiable and continuous. Also, note that the Maclaurin series is just the Taylor series about the point .
The Taylor series of any function about the point is given by the following expression:
If we put , then we will obtain the Maclaurin series.
On simplifying, we get
which is the Maclaurin series.
To calculate the Maclaurin series, we need to calculate the first, second, third up to
Complete step by step answer:
We have to find the Maclaurin series of
Let
As we know that the Maclaurin series is given by
Putting
On simplifying, we get
As we know, from the product rule of differentiation,
Differentiating
On simplification, we get
Putting
Putting the value of
Now, differentiating
On simplification, we get
On simplification, we get
Putting
Now, differentiating
On simplification, we get
Putting
From
On simplification, we get
Therefore, the Maclaurin series for
Note:
To find the Taylor series and Maclaurin series, the function must be infinitely times differentiable and continuous. Also, note that the Maclaurin series is just the Taylor series about the point
The Taylor series of any function
If we put
On simplifying, we get
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