How do you find the explicit formula for a power series?

How do you find the explicit formula for a power series?

Find Explicit Formula for Power Series

1. I took derivative of the power series and got. (−1)n∑∞n=0(xn) The explicit formula would be (−1)n 11−x.
2. I integrated the power series back to the original form and integrated the explicit formula as well. The explicit formula I got was −(−1)n ln(|1-x|) I believe this is incorrect.

What is a series solution?

In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.

How to solve differential equations?

Put the differential equation in the correct initial form,(1).

Find the integrating factor,μ(t),using (10).

Multiply everything in the differential equation by μ(t) and verify that the left side becomes the product rule (μ(t)y(t)) ′ and write it as such.

Integrate both sides,make sure you properly deal with the constant of integration.

Solve for the solution y(t).

What is a linear first order differential equation?

A first order ordinary differential equation is linear if it can be written in the form. y′ + p(t) y = g(t) where p and g are arbitrary functions of t. This is called the standard or canonical form of the first order linear equation. We’ll start by attempting to solve a couple of very simple equations of such type.

What is the second order differential equation?

Second-Order Linear Equations. The order of a differential equation is the order of the highest derivative appearing in the equation. Thus, a second‐order differential equation is one that involves the second derivative of the unknown function but no higher derivatives.