They are the coefficients of the series, with c0 being the constant term, c1 the coefficient of x, c2 the coefficient of x2, and so forth.aero_zeppelin said:Homework Statement
Represent 5x / (10 + x) as a power series, find c0, c1, c2, c3 and c4, find the radius of convergence
The Attempt at a Solution
I think I got the representation fine:
Ʃ from 0 to ∞ = 1/2 [ (-1^n)(x^n+1) / 10^n ]
Radius of Conv. = 10
But what the hell are those c0, c1 etc...? I thought they were the coefficients of the series but I keep getting them wrong...
aero_zeppelin said:My answers are: c0 = 1/2 , c1 = -1/20, c2 = 1/200 ...
Any help please?
aero_zeppelin said:It's been driving me crazy haha. Ok, so it should be:
c0 = ?
c1 = 1/2
c2 = -1/20
c3 = 1/200
c4 = -1/2000
So you just plug in "n" and see the respective coefficients? What happens with c0?
Thanks a lot
A power series representation of a function is a way to express a function as an infinite sum of terms, each of which is a multiple of a variable raised to a non-negative integer power. It is a useful tool in mathematics for approximating and manipulating functions.
To find the power series representation of a function, you can use the Taylor series, which is a specific type of power series that is centered around a specific point. The coefficients of the Taylor series can be found by taking derivatives of the function at that point and plugging them into a formula.
Power series representations are used in many areas of mathematics and science, such as in calculus, physics, and engineering. They are particularly useful for approximating complicated functions and solving differential equations.
No, not all functions can be represented as a power series. A function must be analytic, meaning it can be represented by a convergent power series, for it to have a power series representation. Functions with discontinuities or infinite oscillations cannot be represented as a power series.
One advantage of using a power series representation is that it allows for easy approximation of complicated functions. Power series can also be manipulated algebraically, making it easier to solve differential equations. Additionally, power series can be used to find the behavior of a function near a specific point, which can be useful for understanding the behavior of a function as a whole.