Question

f(x)=x² ln(6+5x) If ∑ n=0[infinity] c nx n is the power series representation of ∫f(x)dx then c 1=c 2= c 3= If n≥4,c n= The radius of convergence is R= If the interval of convergence of the series has end points a​Among the following statements identify the ones which are true: enter ' 1 ' if the statement is true; enter ' 0 ' if it is false At x=a, the series converges. At x=b, the series converges.

Answer 1

To find the power series representation of the integral of f(x), we need to integrate each term individually and express it as a power series. The power series representation of ∫f(x)dx is (1/3)x³. For n≥4, cn=0. The radius of convergence, R, depends on the behavior of ln(6+5x) as x approaches the endpoints of the interval of convergence.

Step-by-step explanation:

In order to find the power series representation of the integral of f(x), we need to integrate each term of the function f(x) individually and then express it as a power series. Let's go step by step:

1. The integral of x² is (1/3)x³.

2. The integral of ln(6+5x) can't be expressed as a power series.

Therefore, the power series representation of ∫f(x)dx is (1/3)x³.

For n≥4, cn = 0 because there are no terms with higher degrees of x.

The radius of convergence, R, depends on the function ln(6+5x). We can determine it by analyzing the behavior of ln(6+5x) as x approaches the endpoints of the interval of convergence.