Power series. tell me if i'm on the right track.

In summary, a power series is an infinite sum of terms with a variable raised to different powers, often used in mathematics to represent functions. It can be written in the form of ∑<sub>n=0</sub>∞ a<sub>n</sub>(x-c)<sup>n</sup>, where a<sub>n</sub> represents the coefficients, x is the variable, and c is the center of the series. A power series will converge if the limit of its terms approaches zero as n approaches infinity, determined by the ratio test. The radius of convergence is the distance from the center of the series to the nearest point where the series converges, and can be calculated using the ratio test. Power series can be
  • #1
crazyformath2
14
0
I have to find the power series f(x)= 3 / (2 -5x).

I devided everything by 2 so I have (1/2) / (1 - 5x/2) = [tex]\sum[/tex] ar^n

and then through some steps I have 15x^n / 2^n+1

and I on the right track? How do I find the interval of convergence?
 
Physics news on Phys.org
  • #2
[tex]f(x)=\frac{3}{2-5x}=\frac{3}{2}*\frac{1}{1-\frac{5x}{2}}=\frac{3}{2}\sum_{n=0}^{\infty}[\frac{5}{2}x]^n[/tex]

To find the radius of convergence u might want to use the ratio test.
 
Last edited:
  • #3
shouldnt that 5/2 x be negative though? does that make a difference?
 

FAQ: Power series. tell me if i'm on the right track.

What is a power series?

A power series is an infinite sum of terms with a variable raised to different powers. It is often used in mathematics to represent functions and can be written in the form of ∑n=0∞ an(x-c)n, where an represents the coefficients, x is the variable, and c is the center of the series.

How do power series converge?

A power series will converge if the limit of its terms approaches zero as n approaches infinity. This condition is known as the ratio test, which states that if the absolute value of (an+1/an) is less than one, the series will converge.

What is the radius of convergence for a power series?

The radius of convergence is the distance from the center of the series to the nearest point where the series converges. It can be calculated using the ratio test, and if the limit of the ratio is equal to one, the radius of convergence is infinite.

Can a power series be used to approximate functions?

Yes, power series can be used to approximate functions by truncating the series after a finite number of terms. The accuracy of the approximation depends on the radius of convergence and the number of terms used. The more terms included, the closer the approximation will be to the actual function.

Can power series be used to solve differential equations?

Yes, power series can be used to solve differential equations by representing the solution as a power series and then finding the coefficients that satisfy the equation. This method is known as the method of Frobenius and is commonly used in physics and engineering.

Similar threads

How to show that these sequences are summable?
Replies
1
Views
1K
Interval of convergence of power series from ratio test
Replies
2
Views
855
On the ratio test for power series
Replies
2
Views
1K
On the definition of radius of convergence; a small supremum technicality
Replies
7
Views
1K
Finding 2 solutions of this Bessel's function using a power series
Replies
8
Views
2K
Pointwise convergence for all real x
Replies
5
Views
495
Power Series and Convergence for ln(x+1)
Replies
22
Views
4K
Possible Values of x for Convergence of Power Series
Replies
1
Views
1K
Intervals of Convergence- Power Series
Replies
11
Views
2K
Solving a Gaussian integral using a power series?
Replies
9
Views
2K

Hot Threads

Recent Insights

Back
Top