Finding the Radius & Interval of Convergence of a Power Series

In summary: So when you write the numerator with a 1, it's telling you to take the 1+1/2 and put it in front of the 1 in the denominator. Not at all. If a_n is the coefficient, then a_1=1, a_2=1+1/2, a_3=1+1/2+1/3. So what's a_3/a_2? 1+1/2. So when you write the numerator with a 1, it's telling you to take the 1+1/2 and put it in front of the 1 in the denominator.
  • #1
craig16
21
0

Homework Statement



[tex]\sum[/tex] from n=1 to inf (1+ 1/2 + ... 1/n)x^n

Find the radius of convergence and the interval of convergence of the given power series.

Homework Equations



Dunno..

The Attempt at a Solution



Stuck thinking about it. I'm not sure if I can combine what's in brackets with the x^n or if I need to do something else.
 
Physics news on Phys.org
  • #2
Try a ratio test.
 
  • #3
Yeah,

Do you think that I should rewrite ( 1 + 1/2 + ... 1/n) as 1/n and do a ration test on
[tex]\sum[/tex] x^n / n

Or is there a special way to treat the first expression in brackets.
 
  • #4
Hmm, I think you just do the sum of a sum..

But if you do that and use the ratio test you get the limit to be zero, and then it would be convergent for all values of x. I just have no clue if that's how it's done.
 
  • #5
craig16 said:
Yeah,

Do you think that I should rewrite ( 1 + 1/2 + ... 1/n) as 1/n and do a ration test on
[tex]\sum[/tex] x^n / n

Or is there a special way to treat the first expression in brackets.

But that's not true. The ratio test expression is (1+1/2+...+1/n+1/(n+1))/(1+1/2+...+1/n). Can you show that the limit of that is 1?
 
  • #6
Hmm,

But how would showing the sum in brackets goes to one with the ratio test help? If that limit equals one the ratio test would be inconclusive.

If I treat this as two sums the (1/n) sum would diverge. So I still don't know what to do.
 
  • #7
craig16 said:
Hmm,

But how would showing the sum in brackets goes to one with the ratio test help? If that limit equals one the ratio test would be inconclusive.

If I treat this as two sums the (1/n) sum would diverge. So I still don't know what to do.

I omitted the x^(n+1)/x^n part of the ratio test. If you can show the limit of what I gave you is 1, then the ratio test will give you x for the ratio.
 
  • #8
Okay, makes sense.

honestly though I don't quite understand the algebra of that limit

n+ 1 / n

you would have, (1/2 + 1/3 ... 1/n+1 ) / (1+1/2... 1/n)

Everything cancels out but ( 1/ n+1 ) / 1 ) ?

Or does the 1/n stay on the bottom
 
  • #9
craig16 said:
Okay, makes sense.

honestly though I don't quite understand the algebra of that limit

n+ 1 / n

you would have, (1/2 + 1/3 ... 1/n+1 ) / (1+1/2... 1/n)

Everything cancels out but ( 1/ n+1 ) / 1 ) ?

Or does the 1/n stay on the bottom

Break it up as (1+1/2+...+1/n)+1/(n+1) in the numerator and (1+1/2+...+1/n) in the denominator. Divide through by the denominator. You get 1+something. What's the something? Can you show the something goes to zero as n->infinity?
 
  • #10
Okay,

I'm curious though why you wrote the numerator with a 1 in it. Since it is n+1 won't it start at 1/2 and finish with 1/n+1, and the denominator start as 1 and finish with 1/n?
 
  • #11
craig16 said:
Okay,

I'm curious though why you wrote the numerator with a 1 in it. Since it is n+1 won't it start at 1/2 and finish with 1/n+1, and the denominator start as 1 and finish with 1/n?

Not at all. If a_n is the coefficient, then a_1=1, a_2=1+1/2, a_3=1+1/2+1/3. So what's a_3/a_2?
 

What is a power series?

A power series is an infinite series of the form f(x) = ∑ an(x-c)n, where an represents the coefficients, c is the center of the series, and n is a non-negative integer. The power series represents a function that is a polynomial with an infinite number of terms.

How do you find the radius of convergence of a power series?

The radius of convergence of a power series is the distance from the center of the series to the point where the series converges. To find the radius of convergence, you can use the ratio test, where the series converges if the limit of |an+1|/|an| is less than 1. The radius of convergence is equal to the limit of this expression as n approaches infinity.

What is the interval of convergence of a power series?

The interval of convergence is the set of values for which the power series converges. The interval of convergence can be determined by testing the endpoints of the interval and checking if the series converges or diverges at those points. The interval may be open, closed, or half-open, depending on the behavior of the series at the endpoints.

Why is it important to find the radius and interval of convergence of a power series?

Knowing the radius and interval of convergence allows us to determine the domain of the function represented by the power series. This is important in applications where we need to know the range of values for which the series accurately represents the function. It also helps us determine the convergence or divergence of the series at specific points.

What are some common methods for finding the radius and interval of convergence?

In addition to the ratio test, other common methods for finding the radius and interval of convergence include the root test, which compares the absolute value of the terms in the series to a geometric series, and the comparison test, which compares the series to another known series with known convergence properties. Computer software, such as Wolfram Alpha, can also be used to find the radius and interval of convergence for a given power series.

Similar threads

Interval of convergence of power series from ratio test
    • Calculus and Beyond Homework Help
Replies
2
Views
166
On the definition of radius of convergence; a small supremum technicality
    • Calculus and Beyond Homework Help
Replies
7
Views
694
On the ratio test for power series
    • Calculus and Beyond Homework Help
Replies
2
Views
700
Intervals of Convergence- Power Series
    • Calculus and Beyond Homework Help
Replies
11
Views
2K
How to show that these sequences are summable?
    • Calculus and Beyond Homework Help
Replies
1
Views
226
Convergence of a Series: Radius and Endpoints
    • Calculus and Beyond Homework Help
Replies
10
Views
920
Interval of uniform convergence of a series
    • Calculus and Beyond Homework Help
Replies
24
Views
2K
Addition of power series and radius of convergence
    • Calculus and Beyond Homework Help
Replies
14
Views
1K
Radius of Convergence and Power Series Expansion Around Multiple Values
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Find the interval of convergence of this power series
    • Calculus and Beyond Homework Help
Replies
2
Views
943

Hot Threads

Recent Insights

Back
Top