Chapter 6 Problems

Thirty chapter problems: 5 from the original book and 25 added for more practice.

Read the chapter problems → · Chapter solutions →

1. Original problem

   What is the definition of a power series?

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2. Original problem

   What is the definition of a power series?

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3. Original problem · clarified

   Prove that $f\left(x\right)g\left(x\right)={\sum }_{n=0}^{\infty }\left({\sum }_{i=0}^n{a}_i{b}_{n-i}\right)(x-c{)}^n$.

   Clarified version: the inner sum ends at n, so it includes exactly the coefficient pairs whose indices add to n.

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4. Original problem · clarified

   For $f\left(x\right)={\sum }_{n=0}^{\infty }{n}^2{x}^n$ and $g\left(x\right)={\sum }_{n=0}^{\infty }(2n+1)x^n$ compute the following.

   1. $f\left(x\right)+g\left(x\right)$
   2. $f\left(x\right)-g\left(x\right)$
   3. $\mathrm{df}/\mathrm{dx}$
   4. $\int g\left(x\right)\mathrm{dx}$

   Clarified version: g(x) includes the factor xⁿ in each term, making it a power series in x.

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5. Original problem

   Taylor expand the following function around the given value and note which are Maclaurin Series.

   1. 
   2. 
   3. 
   4. 
   5. 
   6. 
   7. 

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6. Added problem

   Find the radius and interval of convergence of ${\sum }_{n=0}^{\infty }{x}^n$.

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7. Added problem

   Find the center, radius, and interval of convergence of ${\sum }_{n=0}^{\infty }(x-2{)}^n/3^n$.

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8. Added problem

   Find the radius and interval of convergence of ${\sum }_{n=1}^{\infty }{x}^n/n$.

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9. Added problem

   Find the radius and interval of convergence of ${\sum }_{n=1}^{\infty }{x}^n/n^2$.

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10. Added problem

    Find the radius of convergence of ${\sum }_{n=0}^{\infty }{x}^n/n!$.

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11. Added problem

    For which real x does ${\sum }_{n=0}^{\infty }n!x^n$ converge?

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12. Added problem

    Differentiate the geometric power series to represent $1/(1-x{)}^2$. State the interval of convergence.

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13. Added problem

    Integrate ${\sum }_{n=0}^{\infty }{x}^n$ from 0 to x and state the resulting function and interval.

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14. Added problem

    Find the cubic Taylor polynomial for $e^x$ centered at 1.

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15. Added problem

    Find the cubic Taylor polynomial of $\ln x$ centered at 2.

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16. Added problem

    Taylor expand $x^3$ centered at 1.

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17. Added problem

    Taylor expand $1/x$ centered at 2 and state where the series converges.

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18. Added problem

    Taylor expand $\sin x$ centered at $\pi /2$.

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19. Added problem

    A Taylor series starts $5-2(x-3)+4(x-3{)}^2+\dots$. Find $f\prime \left(3\right)$ and $f″\left(3\right)$.

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20. Added problem

    Use the cubic Maclaurin polynomial to approximate $e^{0.1}$ and bound the error with Taylor’s theorem.

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21. Added problem

    How is a Maclaurin series related to a Taylor series?

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22. Added problem

    Find the Maclaurin series for $e^{2x}$ and its interval of convergence.

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23. Added problem

    Find the Maclaurin series for $\cos \left(3x\right)$.

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24. Added problem

    Find the Maclaurin series and interval of convergence of $\ln (1+2x)$.

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25. Added problem

    Give the first three nonzero terms of the Maclaurin series of $\sin \left(x^2\right)$.

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26. Added problem

    Find the cubic Maclaurin polynomial of $e^x\sin x$.

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27. Added problem

    Find the Maclaurin series of $1/(1+x^2)$ and its interval of convergence.

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28. Added problem

    What is the coefficient of x⁵ in the Maclaurin series for $x^2{e}^x$?

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29. Added problem

    Must a Taylor series represent its function at every point in the function’s domain? Give a concrete example.

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30. Added problem

    Find the quadratic Maclaurin polynomial for $\mathrm{sqrt}(1+x)$.

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