Explicit formula for a convergent series

In summary, the conversation is about finding the radius of convergence and explicit formula for the function f(x) = 1 + 2x + x2 + 2x3 + x4 + 2x5 + x6... The attempt at a solution involves using a formula similar to xn2c, with c being dependent on n and equal to 1 when n is odd and 0 when n is 0 or even. However, the person is having difficulty finding the correct value for c. The solution involves rewriting the function as (1+2x)+(1+2x)x^2+(1+2x)x^4+... in order to see what was previously missing.
  • #1
trulyfalse
35
0
Hello PF.

Homework Statement


A function f is defined by f(x) = 1 + 2x + x2 + 2x3 + x4 + 2x5 + x6... Find the radius of convergence of the series and the explicit formula for f(x).

Homework Equations


The Attempt at a Solution


I know that the formula for the series is going to be similar to the function xn2c, where c is some expression dependent on n that is equal to 1 when n is odd and equal to 0 when n is 0 or when n is even. However, I'm having difficulty finding what c actually is in terms of n. Maybe there's something obvious that I'm not seeing here?
 
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  • #2
trulyfalse said:
Hello PF.

Homework Statement


A function f is defined by f(x) = 1 + 2x + x2 + 2x3 + x4 + 2x5 + x6... Find the radius of convergence of the series and the explicit formula for f(x).


Homework Equations





The Attempt at a Solution


I know that the formula for the series is going to be similar to the function xn2c, where c is some expression dependent on n that is equal to 1 when n is odd and equal to 0 when n is 0 or when n is even. However, I'm having difficulty finding what c actually is in terms of n. Maybe there's something obvious that I'm not seeing here?

Write it as (1+2x)+(1+2x)x^2+(1+2x)x^4+...Now do you see what you are missing?
 

Related to Explicit formula for a convergent series

1. What is an explicit formula for a convergent series?

An explicit formula for a convergent series is a mathematical expression that can be used to calculate the sum of all terms in a convergent series, or a series that has a finite sum. It is also known as a closed-form formula or a closed-form expression.

2. How is an explicit formula different from a recursive formula?

An explicit formula is a direct way to calculate the sum of a series, while a recursive formula requires the use of previous terms in the series to calculate the next term. An explicit formula is generally more efficient and easier to use than a recursive formula.

3. What are the benefits of using an explicit formula for a convergent series?

Using an explicit formula allows for a quick and accurate calculation of the sum of a convergent series. It also provides a deeper understanding of the behavior and patterns within the series, which can be useful in further mathematical analysis and applications.

4. Can an explicit formula be used for all types of convergent series?

No, an explicit formula can only be used for certain types of convergent series. It is most commonly used for arithmetic and geometric series, but it may not be applicable for more complex series with irregular patterns or infinite terms.

5. How do you determine if a convergent series has an explicit formula?

In some cases, an explicit formula for a convergent series can be derived through mathematical techniques such as algebraic manipulation or using known summation formulas. However, for more complex series, it may not be possible to find an explicit formula and other methods, such as numerical approximations, may be used instead.

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