6.2.1: Review of Power Series (Exercises)

Q6.2.1

1.For each power series find the center, the interval and the radius of convergence.

1. \({\displaystyle \sum_{n=0}^\infty {(-1)^n\over2^nn}(x-1)^n}\)
2. \({\displaystyle \sum_{n=0}^\infty 2^nn(x-2)^n}\)
3. \({\displaystyle \sum_{n=0}^\infty {n!\over9^n}x^n}\)
4. \({\displaystyle \sum_{n=0}^\infty{n(n+1)\over16^n}(x-2)^n}\)
5. \({\displaystyle \sum_{n=0}^\infty (-1)^n{7^n\over n!}x^n}\)
6. \({\displaystyle \sum_{n=0}^\infty {3^n\over4^{n+1}(n+1)^2}(x+7)^n}\)

Q6.2.2

For Exercises 6.2.2-6.2.5 verify that the given power series is a solution of the indicated differential equation.

2. \({\displaystyle \sum_{n=0}^\infty {(-1)^n\over (2n)!}x^{2n}}\), \(y''+y=0\)

3. \({\displaystyle \sum_{n=0}^\infty {1\over2^nn!}x^{2n}}\),  \(y''-xy'-y=0\)

4. \({\displaystyle \sum_{n=0}^\infty {(-1)^n}x^n}\), \((x+1)y''+2y'=0\), \(-1<x<1\)

5. \({\displaystyle \sum_{n=1}^\infty {(-1)^{n+1}\over n}x^n}\), \((x+1)y''+y'=0\), \(-1<x \le 1\)

Q6.2.3

Suppose \(y(x)=\sum_{n=0}^{\infty} a_{n}x^{n}\) on an open interval that contains \(x_0~=~0\). For Exercises 6.2.6-6.2.10  find a power series in x.

6. \((2+x)y''+xy'+3y\)

7. \((1+3x^2)y''+3x^2y'-2y\)

8. \((1+2x^2)y''+(2-3x)y'+4y\)

9. \((1+x^2)y''+(2-x)y'+3y\)

10. \((1+3x^2)y''-2xy'+4y\)

Q6.2.4

11. Suppose \(y(x)=\displaystyle \sum_{n=0}^\infty a_n(x+1)^n\) on an open interval that contains \(x_0~=~-1\). Find a power series in \(x+1\) for \[xy''+(4+2x)y'+(2+x)y.\nonumber \]

12. Suppose \(y(x)=\displaystyle \sum_{n=0}^\infty a_n(x-2)^n\) on an open interval that contains \(x_0~=~2\). Find a power series in \(x-2\) for \[x^2y''+2xy'-3xy.\nonumber \]