Finding the Radius of Convergence for (sum from n=0 to infinity)7^(-n)x^(n)

In summary, the conversation discussed finding the radius of convergence for a given series and the use of the Ratio Test to solve it. After some initial confusion, it was determined that the correct approach is to use 7^{-n-1} instead of 7^{-n+1}. This results in a limit of x/7 and a radius of convergence of 7, as the inverse of the limit must be taken to find the actual radius.
  • #1
lmannoia
32
0

Homework Statement


Find the radius of convergence for (sum from n=0 to infinity)7^(-n)x^(n).


Homework Equations





The Attempt at a Solution


The problem above it was a similar sum, (7^n)(x^n). That answer was that the radius of convergence was 1/7.
To do this one that I posted up there, I tried to use the Ratio Test...
7^(-n+1)x^(n+1) all over (7^-n)(x^n). I ended up getting 1/49, but that's wrong. Any idea of what I'm doing incorrectly, or do I just have the wrong approach to solve this one altogether?
 
Physics news on Phys.org
  • #2
Shouldnt that be 7^{-n-1} instead of 7^{-n+1}?
 
  • #3
Oh wow, I can't believe I made that mistake. Thank you!
 
  • #4
But wait, if you do 7^{-n-1}x^{n+1}/7^{-n}x^{n}, doesn't it get down to x/7? In which case, wouldn't R be 1/7?
 
  • #5
Yes, the limit is 1/7. But to find the convergence radius, you have to invert that number. So the answer would be 7...
 

What is the radius of convergence?

The radius of convergence is a mathematical concept that relates to power series, which are infinite sums of terms that involve powers of a variable. It is the distance from the center of the series at which the series will converge, meaning that the sum of all the terms will approach a finite value as the distance from the center increases.

How is the radius of convergence calculated?

The radius of convergence is calculated using the ratio test, which involves taking the limit of the absolute value of the ratio of consecutive terms in the series. If the limit is less than 1, the series will converge, and the radius of convergence can be found by taking the reciprocal of this limit.

Why is the radius of convergence important?

The radius of convergence is important because it determines the values for which a power series will converge. This allows us to approximate functions with infinite sums and to extend the domain of functions beyond their usual limits.

What happens if the radius of convergence is infinite?

If the radius of convergence is infinite, it means that the series will converge for all values of the variable. This is also known as a convergent power series, and it is a powerful tool for approximating and extending functions.

Can the radius of convergence be negative?

No, the radius of convergence cannot be negative. It represents a distance, and distances cannot be negative. However, the center of the series can be a negative value, which means that the series will converge for all values of the variable that are within that distance from the center.

Similar threads

Interval of convergence of power series from ratio test
    • Calculus and Beyond Homework Help
Replies
2
Views
184
How to show that these sequences are summable?
    • Calculus and Beyond Homework Help
Replies
1
Views
255
On the definition of radius of convergence; a small supremum technicality
    • Calculus and Beyond Homework Help
Replies
7
Views
707
Prove that this series converges uniformly on R
    • Calculus and Beyond Homework Help
Replies
4
Views
306
On the ratio test for power series
    • Calculus and Beyond Homework Help
Replies
2
Views
711
On pointwise convergence of Fourier series
    • Calculus and Beyond Homework Help
Replies
3
Views
414
Addition of power series and radius of convergence
    • Calculus and Beyond Homework Help
Replies
14
Views
1K
Set of convergence of a Power series
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
What is the radius of convergence for a series with logarithmic terms?
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Power series: radius of convergence
    • Calculus and Beyond Homework Help
Replies
8
Views
887

Hot Threads

Recent Insights

Back
Top