Power series calculus pdf

Then, for even values of n , we have. Verify that the expression found in Example 7. As an Amazon Associate we earn from qualifying purchases. Want to cite, share, or modify this book? Skip to Content Go to accessibility page. Calculus Volume 3 7. My highlights. Table of contents. Chapter Review. Answer Key. Substitute the power series expressions into the differential equation.

Re-index sums as necessary to combine terms and simplify the expression. Equate coefficients of like powers of x x to determine values for the coefficients a n a n in the power series. Substitute the coefficients back into the power series and write the solution. Series Solutions to Differential Equations Find a power series solution for the following differential equations.

Find a power series solution for the following differential equations. Power Series Solution to the Bessel Equation Find a power series solution to the Bessel equation of order 0 and graph the solution. Section 7. This number is called the radius of convergence for the series. What happens at these points will not change the radius of convergence. These two concepts are fairly closely tied together. In this case the power series becomes,.

Note that we had to strip out the first term since it was the only non-zero term in the series. From this we can get the radius of convergence and most of the interval of convergence with the possible exception of the endpoints.

With all that said, the best tests to use here are almost always the ratio or root test. The limit is then,. So, we have,. Notice that we now have the radius of convergence for this power series.

These are exactly the conditions required for the radius of convergence. All we need to do is determine if the power series will converge or diverge at the endpoints of this interval.

The way to determine convergence at these points is to simply plug them into the original power series and see if the series converges or diverges using any test necessary. So, in this case the power series will not converge for either endpoint.

The interval of convergence is then,. The power series could converge at either both of the endpoints or only one of the endpoints. We need to be careful here in determining the interval of convergence. In other words, we need to factor a 4 out of the absolute value bars in order to get the correct radius of convergence.

Doing this gives,. So, the power series converges for one of the endpoints, but not the other.