Learn – Calculus – Sequences

Convergence and divergence

Exercises

Problem set

For each of the following problems, give whether the sequence converges or diverges. If the sequence converges, give what the sequence converges to. If the sequence diverges, give if the sequence oscillates back and forth or diverges to \infty.

  1. \left\{\frac{1}{n}\right\}
  2. \left\{\sqrt[3]{n}\right\}
  3. \left\{\frac{1}{\sqrt{n}}\right\}
  4. \left\{-2^n\right\}
  5. \left\{1-\frac{1}{n^2}\right\}
  6. \left\{(-1)^n\right\}
  7. \left\{\cos\left(\frac{n\pi}{2}\right)\right\}

Problem set

For each of the following problems, give whether the sequence converges or diverges. If the sequence converges, give what the sequence converges to. If the sequence diverges, give if the sequence oscillates back and forth or diverges to \infty.

  1. \left\{0.5^n\right\}
  2. \left\{\ln n\right\}
  3. \left\{\frac{(-1)^n}{n}\right\}
  4. \left\{(-0.5)^n\right\}
  5. \left\{3+(-0.9)^n\right\}
  6. \left\{\frac{e^n}{n!}\right\}

Convergence laws

Exercises

Problem set

In each of the following, give, with reason, if the given sequence converges or diverges.

  1. \left\{\frac{3}{n}+0.9^n\right\}_{n=1}^{\infty}

  2. \left\{\frac{9}{n^3}-\frac{7}{n^2}\right\}_{n=1}^{\infty}

  3. \left\{\frac{2}{1}+\frac{3}{\ln 2},\, \frac{2}{2}+\frac{3}{\ln 3},\,\frac{2}{3}+\frac{3}{\ln 4},\,\cdots\right\}

  4. \left\{\frac{4}{1},\, \frac{5}{2},\, \frac{6}{3},\, \cdots \right\}

  5. \left\{\frac{3n^2+4n-5}{5n^2+9}\right\}_{n=1}^{\infty}

Using functions of real variables

Exercises

Problem set

In each of the following, the limit of a sequence is given in the limit notation. Evaluate the limit.

  1. \lim\limits_{n\to\infty}\sin\left(\frac{\frac{n\pi}{2}+3}{n}\right)

  2. \lim\limits_{n\to\infty}\ln(n+10) - \ln(n+3)

  3. \lim\limits_{n\to\infty}\sin\left(\frac{(-1)^n}{n}\right)

  4. \lim\limits_{n\to\infty}\left\lceil\frac{(-1)^n}{n}\right\rceil. (Note that, \lceil{x}\rceil is the smallest integer that is greater than or equal to x. For example, \lceil{5.1}\rceil = 6,\,\, \lceil{4.9}\rceil = 5,\,\,\lceil{5}\rceil = 5. And, \lfloor{x}\rfloor is the greatest integer that is less than or equal to x. For example, \lfloor{5.1}\rfloor = 5,\,\, \lfloor{4.9}\rfloor = 4,\,\,\lfloor{5}\rfloor = 5.)

Problem set

In each of the following, the limit of a sequence is given in the limit notation. Evaluate the limit.

  1. \lim\limits_{n\to\infty}\frac{\ln n}{n}

  2. \lim\limits_{n\to\infty}\frac{(\ln n)^2}{\sqrt{n}}

  3. \lim\limits_{n\to \infty}n\sin\left(\frac{10}{n}\right)

  4. \lim\limits_{n\to \infty}n\tan\left(\frac{7}{n}\right)

  5. \lim\limits_{n\to \infty}\frac{\sin\left(\frac{10}{n}\right)}{\tan\left(\frac{9}{n}\right)}

  6. \lim\limits_{n\to \infty}\sqrt{\frac{\sin\left(\frac{50}{n}\right)}{\tan\left(\frac{2}{n}\right)}}

Problem set

In each of the following, the limit of a sequence is given in the limit notation. Evaluate the limit.

  1. \lim\limits_{n\to\infty} 10^{1/n}

  2. \lim\limits_{n\to\infty} 0.1^{1/n}

  3. \lim\limits_{n\to\infty} \frac{n^5}{2^n}

  4. \lim\limits_{n\to\infty} \frac{n^{9999}}{e^n}

  5. \lim\limits_{n\to\infty} \frac{e^n}{n!}

  6. \lim\limits_{n\to\infty} \frac{9999^n}{n!}

  7. \lim\limits_{n\to\infty} \sqrt[n]{n}

  8. \lim\limits_{n\to\infty} \left(1+\frac{k}{n}\right)^n