Chapter 3 Problems

Thirty chapter problems: 6 from the original book and 24 added for more practice.

Read the chapter problems → · Chapter solutions →

1. Original problem · clarified

   Determine the type of each of the following series.

   1. Clarified pattern: $\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\dots$ (terms $\frac{(-1{)}^n}{n}$ for $n\ge 2$).
   2. 
   3. 
   4. 
   5. 

   Clarification for part (a): this web edition explicitly uses 1/2−1/3+1/4−… . The source’s displayed 1/2−1/3+1/3−… does not specify an unambiguous continuation. Parts (b)–(e) retain their source expressions.

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2. Original problem

   Determine if the following series are convergent or divergent.

   1. 
   2. 

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3. Original problem

   Use partial fraction decomposition to determine if the following series is telescopic.

   1. 

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4. Original problem

   Determine the value of the following series to the 30th term.

   1. 
   2. 

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5. Original problem

   Determine the value of the following series to an infinite number of terms.

   

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6. Original problem

   Create Fourier Series for the following periodic functions.

   1. 
   2. 

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7. Added problem

   Find the sum of $6-2+\frac{2}{3}-\frac{2}{9}+\dots$.

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8. Added problem

   For ${\sum }_{n=0}^{\infty }5(\frac{2}{5}{)}^n$, find the sum of its first four terms and its infinite sum.

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9. Added problem

   Evaluate ${\sum }_{n=0}^{\infty }(\frac{1}{\sqrt{2}}{)}^n$.

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10. Added problem

    Find the first-N-term sum of ${\sum }_{n=1}^{\infty }3(-2{)}^{n-1}$ and determine whether the infinite series converges.

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11. Added problem

    Evaluate ${\sum }_{n=3}^{\infty }\frac{4}{2^n}$.

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12. Added problem

    Use a geometric series to express $0.272727\dots$ as a reduced fraction.

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13. Added problem

    Evaluate ${\sum }_{n=1}^{\infty }\frac{1}{n(n+2)}$ by telescoping.

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14. Added problem

    Evaluate ${\sum }_{n=1}^{\infty }(\frac{1}{n+2}-\frac{1}{n+5})$.

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15. Added problem

    Find the partial sums and determine convergence of ${\sum }_{n=1}^{\infty }\ln \left(\frac{n+1}{n}\right)$.

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16. Added problem

    Evaluate ${\sum }_{n=1}^{\infty }(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}})$.

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17. Added problem

    What is the smallest number of terms needed to approximate ${\sum }_{n=1}^{\infty }\frac{1}{n(n+1)}$ with absolute error strictly less than $0.001$?

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18. Added problem

    Evaluate ${\sum }_{n=1}^{\infty }(\frac{2}{n}-\frac{2}{n+1})$.

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19. Added problem

    Prove that ${\sum }_{n=1}^{\infty }\frac{1}{n}$ diverges by grouping terms in blocks whose endpoints are powers of 2.

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20. Added problem

    Determine whether ${\sum }_{n=1}^{\infty }\frac{1}{2n-1}$ converges. Give a comparison with the harmonic series.

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21. Added problem

    Show that the alternating harmonic series $1-\frac{1}{2}+\frac{1}{3}-\dots$ sums to $\ln 2$ by integrating a finite geometric identity.

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22. Added problem

    Evaluate the paired series ${\sum }_{n=1}^{\infty }(\frac{1}{n}-\frac{1}{n+2})$ and explain why its finite partial sums matter.

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23. Added problem

    Find the real Fourier series of the $2\pi$-periodic function $f\left(x\right)=1$.

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24. Added problem

    Find the Fourier series of the $2\pi$-periodic extension of $f\left(x\right)=x$ on , including its value at the periodic jumps.

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25. Added problem

    Find the Fourier series of the $2\pi$-periodic extension of $f\left(x\right)=\left|x\right|$ on $-\pi \le x\le \pi$.

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26. Added problem

    Find the Fourier series of the $2\pi$-periodic function equal to $-1$ for and $1$ for . State the sum at each jump.

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27. Added problem

    Find the Fourier series of the $2\pi$-periodic extension of $f\left(x\right)=x^2$ on $-\pi \le x\le \pi$, including the endpoint sum.

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28. Added problem

    List the real Fourier coefficients of $f\left(x\right)=3\cos \left(2x\right)-2\sin \left(3x\right)$ in the convention $f=\frac{a_0}{2}+{\sum }_{n=1}^{\infty }(a_n\cos nx+b_n\sin nx)$.

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29. Added problem

    Find the Fourier series of the 4-periodic extension of $f\left(x\right)=x$ on , and state its sum at the jumps.

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30. Added problem

    A square wave has Fourier series $\frac{4}{\pi }{\sum }_{k=0}^{\infty }\frac{\sin \left(\right(2k+1\left)x\right)}{2k+1}$. Write its first three nonzero terms and evaluate that approximation at $x=\frac{\pi }{2}$.

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