BvU said:What is the power series for 1/(1-x) ?
BvU said:Good. Perfect, in fact. If you want you can rewrite it in a more conventional form (##\ \ \displaystyle\sum_{n=0}^\infty \ \ ##)
A power series is an infinite series of the form ∑n=0∞ cn(x-a)n, where cn are constants, x is a variable, and a is a fixed point.
To differentiate a power series, you can use the power rule, which states that the derivative of xn is nxn-1. You can apply this rule to each term in the series to get the derivative of the entire series.
The radius of convergence of a power series is the distance from the center point a to the nearest point where the series still converges. It is given by the formula R = 1/limn→∞ |cn+1|/|cn|. If the limit does not exist, the radius of convergence is infinite.
No, a power series can only converge within its radius of convergence. At the endpoints, the series may or may not converge, depending on the specific values of x. This is why it is important to check the convergence at the boundaries when using a power series to represent a function.
Differentiating a power series allows us to find the derivative of a function represented by the series. This can be useful in finding the derivative of functions that are difficult to differentiate using traditional methods. It also allows us to approximate the values of the derivative at different points within the radius of convergence.