Radius of Convergence for Series with (-1)^(n-1) and x^n Terms

In summary: The ratio test gives "L = lim_{n->infty} |a_{n+1}/a_n|". For the given series, a_n = x^n/n. Then a_{n+1} = x^{n+1}/(n+1). Find the limit of the ratio, then set it less than 1 and solve for x. The series converges for "-1 < x <= 1". The series converges to "f(x) = ln(1+x)" when "-1 < x < 1" and to "f(x) = ln(2)" when "x = 1". I hope this helps.In summary,
  • #1
tracedinair
50
0

Homework Statement



Find the radius of convergence of [tex]\sum[/tex] n=1 to ∞ of [(-1)^(n-1) x^(n)/(n)] and give a formula for the value of the series at the right hand endpoint.

Homework Equations




The Attempt at a Solution



Not really sure how to start this. I know I'm supposed to use the root test or the ratio test and then probably Abel's Thm for series. but the n-1 in the series is throwing me off. I've never seen that before. Any help is appreciated.
 
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  • #2
Your series is just f(x)=x-x^2/2+x^3/3+... Just use the ratio test to find the radius of convergence. To find the actual value at the endpoint you could just ask yourself is you have seen that kind of series before. If it doesn't look familiar, try finding f'(x) and see if that series looks familiar.
 
  • #3
Dick said:
Your series is just f(x)=x-x^2/2+x^3/3+... Just use the ratio test to find the radius of convergence. To find the actual value at the endpoint you could just ask yourself is you have seen that kind of series before. If it doesn't look familiar, try finding f'(x) and see if that series looks familiar.

So I took f'(x) = 1 - x + x^2 - x^3 + ... so, the endpoint is going to look like x^(n+1)? I need a formula for the value at the right hand endpoint..would that work?
 
  • #4
No, you have to recognize the series, my point to looking at f'(x) was that it's geometric, it's 1/(1+x). if that's f'(x), what's f(x)?
 
  • #5
If f'(x) = 1/(1+x) then f(x) = ln(x+1)?
 
  • #6
ln(x+1)+C, you mean. You have to fix the constant C by finding f(0). But yes, that's the idea.
 
  • #7
Alright, thank you.
 
  • #8
tracedinair said:

Homework Statement



Find the radius of convergence of [tex]\sum[/tex] n=1 to ∞ of [(-1)^(n-1) x^(n)/(n)] and give a formula for the value of the series at the right hand endpoint.

Homework Equations




The Attempt at a Solution



Not really sure how to start this. I know I'm supposed to use the root test or the ratio test and then probably Abel's Thm for series. but the n-1 in the series is throwing me off. I've never seen that before. Any help is appreciated.
The ratio test applies to a positive series and a power series converges absolutely inside its "radius of convergence".

Take the absolute value (drop the "[itex](-1)^{n-1}[/itex]") and then use the ratio test.
 

1. What is the radius of convergence?

The radius of convergence is a mathematical concept that is used in power series to determine the values of the independent variable for which the series will converge. It is represented by the letter R and is defined as the distance from the center of the power series to the nearest point where the series converges.

2. How is the radius of convergence calculated?

The radius of convergence can be calculated using the ratio test, which involves taking the limit of the absolute value of the ratio of consecutive terms in the series. If this limit is less than 1, the series converges and the radius of convergence is equal to the reciprocal of the limit. If the limit is greater than 1, the series diverges and the radius of convergence is 0. If the limit is equal to 1, further tests are needed to determine the convergence of the series.

3. What is the significance of the radius of convergence?

The radius of convergence is important because it tells us the range of values for the independent variable where the power series will converge. This allows us to determine the interval of convergence for the series and use it to approximate functions or solve differential equations.

4. Can the radius of convergence be negative?

No, the radius of convergence cannot be negative. It is always a positive value or 0. The negative sign in the notation for the radius of convergence, R, indicates that it is a distance and not a direction.

5. How does the radius of convergence relate to the convergence of a power series?

The radius of convergence is directly related to the convergence of a power series. If the value of the independent variable is within the radius of convergence, the series will converge. If it is outside the radius of convergence, the series will diverge. Additionally, the radius of convergence can tell us about the behavior of the series at the endpoints of the interval of convergence.

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