Help with power series representation

In summary, the conversation discusses finding a power series representation and determining the radius of convergence for the function f(t) = ln(2-t). The solution involves taking the derivative of ln(2-t) and transforming the integral of 1/(2-t) into a more recognizable form using a power series. The final answer is a power series representation of (\frac{t}{2})^{n} with a radius of convergence of 2.
  • #1
bobbarkernar
48
0

Homework Statement



find a power series representation for the function and determine the radius of convergence.

f(t)= ln(2-t)


Homework Equations





The Attempt at a Solution



i first took the derivative of ln(2-t) which is 1/(t-2)

then i tried to write the integral 1/(t-2) in the form of a power series i got the sum for n=0 to infinity of (t/2)^n i don't know if this is write if someone can please help me
 
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  • #2
If [itex] f(t) = \ln(2-t) [/tex] then write this as follows:[itex] -\ln(2-t) = \int \frac{1}{2-t} \; dt [/tex]

How would you transform [itex] \frac{1}{2-t} [/tex] into a more recognizable form: i.e. [itex] \frac{1}{1-t} [/itex]?
 
  • #3
i would factor out a 1/2 so i would have (1/2)*1/(1-(1/2t))
 
  • #4
ok then what would you do?
 
  • #5
i would write that as a power series:
(1/2)*the sum for n=0 to infinity of (1/2t)^n
 
  • #6
correct you have:

[tex] \frac{1}{2}\sum_{n=0}^{\infty} (\frac{t}{2})^{n} [/tex]
 
  • #7
so is that the answer??
 
  • #8
yes that is the answer.

or write it like this:

[tex] \sum_{n=0}^{\infty} \left(\frac{t^{n}}{2^{n+1}}\right) [/tex]
 
  • #9
ok thank you very much
 
  • #10
courtrigrad said:
correct you have:

[tex] \frac{1}{2}\sum_{n=0}^{\infty} (\frac{t}{2})^{n} [/tex]


I'm confused... What happens to the integral?
 

Related to Help with power series representation

1. What is a power series representation?

A power series representation is a mathematical expression that represents a function as an infinite sum of terms, each with an increasing power of a variable. It is often used to approximate functions and solve problems in calculus and other branches of mathematics.

2. How do you find the power series representation of a function?

To find the power series representation of a function, you can use the Taylor series expansion or the Maclaurin series expansion. These methods involve finding the derivatives of the function at a specific point and plugging them into a formula to determine the coefficients of the power series.

3. What is the purpose of using a power series representation?

The purpose of using a power series representation is to approximate a function in terms of simpler functions, such as polynomials, which are easier to work with. This allows for easier computation and analysis of the function, especially when the function is difficult to evaluate directly.

4. Are there any limitations to using a power series representation?

Yes, there are limitations to using a power series representation. It is only applicable to functions that can be expressed as an infinite sum of terms with increasing powers of a variable. It also may not accurately represent the function for all values of the variable, as it is an approximation.

5. Can a power series representation be used to find the value of a function at a specific point?

Yes, a power series representation can be used to find the value of a function at a specific point. This is known as the convergence of a power series, and it can be determined by finding the radius of convergence. If the value of the variable is within this radius, the power series will converge and give an accurate approximation of the function at that point.

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