How Do You Evaluate the Indefinite Integral as a Power Series?

In summary, to evaluate the indefinite integral as a power series, we can expand ln(1-t) as a MacLauren series and substitute it into the given equation, then find the first five non-zero terms of the power series representation centered at t=0. The radius of convergence is 1 and a constant "C" should be included.
  • #1
ajkess1994
9
0
Evaluate the indefinite integral as a power series
∫[ln(1−t)/7t]dt.
Find the first five non-zero terms of power series representation centered at t=0.

Answer: f(t)=

What is the radius of convergence?
Answer: R= 1

Note: Remember to include a constant "C".

This problem has been difficult for me for awhile now. I have my radius but I can't figure out my five f(t) terms it's asking for.
Try to build those terms from this:

(-t^(n))/(7n^(2))

Also n=1
 
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  • #2
ajkess1994 said:
Evaluate the indefinite integral as a power series
∫[ln(1−t)/7t]dt.
Find the first five non-zero terms of power series representation centered at t=0.

Answer: f(t)=

What is the radius of convergence?
Answer: R= 1

Note: Remember to include a constant "C".

This problem has been difficult for me for awhile now. I have my radius but I can't figure out my five f(t) terms it's asking for.
Try to build those terms from this:

(-t^(n))/(7n^(2))

Also n=1

Hi ajkess1994,

Can you expand $\ln(1−t)$ as a MacLauren series?
Substitute it, and the rest should follow.
 
  • #3
Yes, Arsen I can expan the ln(1-t) to a MacLauren series, and from there finding the "nth" terms shouldn't be too difficult, thank you for the suggestioon.
 

1. What is a Power series and how does it differ from a Maclauren series?

A Power series is a mathematical series that is used to represent a function as a sum of terms, while a Maclauren series is a specific type of power series that is centered at x=0. The main difference between the two is that a Maclauren series has all of its terms in powers of x, while a general power series can have any variable raised to any power.

2. How is a Power or Maclauren series used in calculus?

Power and Maclauren series are often used in calculus to approximate a function, especially when the function is difficult to integrate or differentiate. By expanding a function into a Maclauren series, we can easily find the derivatives and integrals of the function.

3. What is the formula for a Maclauren series?

The formula for a Maclauren series is f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + ... + f(n)(0)x^n/n! , where f(n)(0) represents the n-th derivative of the function evaluated at x=0.

4. Can a Power or Maclauren series represent any function?

No, a Power or Maclauren series can only represent functions that are infinitely differentiable within a certain interval. Additionally, the series may only converge to the actual value of the function within a certain radius of convergence.

5. How can I determine the radius of convergence for a Power or Maclauren series?

The radius of convergence can be determined using the ratio test, which states that the series will converge for values of x within a certain radius if the limit of the absolute value of the ratio of consecutive terms approaches a finite value. In other words, the series will converge if the limit of |(an+1/an)| as n approaches infinity is less than 1.

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