Learn – Calculus – Series

Geometric series

Exercises

Problem set

For each of the following problems, give whether the given series converges or diverges. If the series converges, give what the series converges to.

  1. \frac{2}{3} + \frac{2}{3^2} + \frac{2}{3^3} + \cdots

  2. \frac{50}{2^3} + \frac{50}{2^4} + \frac{50}{2^5} + \cdots

  3. \sum\limits_{n=1}^{\infty} \left(\frac{1}{3}\right)^{n-1}

  4. \sum\limits_{n=4}^{\infty} \frac{1}{3^{n-1}}

  5. \sum\limits_{n=2}^{\infty} \frac{(-1)^n}{10^n}

  6. \sum\limits_{n=0}^{\infty} \frac{1}{2^{-n}}

  7. \sum\limits_{n=0}^{\infty} (-1.1)^n

Integral test

Exercises

Problem set

For each of the following problems, give, with a justification, whether the given series converges or diverges.

  1. 1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\cdots

  2. 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots

  3. 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+\cdots

Problem set

For each of the following problems, give, with a justification, whether the given series converges or diverges.

  1. \sum\limits_{n=1}^{\infty} \frac{1}{1+n^2}

  2. \sum\limits_{n=1}^{\infty} \frac{1}{(n+2)^2+9}

  3. \sum\limits_{n=1}^{\infty} \frac{3n^2}{n^3+5}

Divergence test

Exercises

Problem set

For each of the following problems, give, with a justification, whether the given series converges or diverges.

  1. \sum\limits_{n=1}^{\infty} n^{1/n}

  2. \sum\limits_{n=1}^{\infty} \sin\left(n^{1/n}\right)

Comparison tests

Exercises

Problem set

For each of the following problems, give, with a justification, whether the given series converges or diverges.

  1. \sum\limits_{n=1}^{\infty} \frac{1}{n^2+3}

  2. \sum\limits_{n=1}^{\infty} \frac{1}{n^2-3}

  3. \sum\limits_{n=1}^{\infty} \frac{1}{\sqrt{n}+3}

  4. \sum\limits_{n=1}^{\infty} \frac{|\sin n|}{n^2}

  5. \sum\limits_{n=1}^{\infty} \frac{1}{\sqrt{n^2+999}}

  6. \sum\limits_{n=1}^{\infty} \frac{n}{\sqrt{n^3+100}}

Problem set

For each of the following problems, give, with a justification, whether the given series converges or diverges.

  1. \sum\limits_{n=1}^{\infty} \frac{3\sqrt{n}+5}{2n^2-3}

  2. \sum\limits_{n=1}^{\infty} \frac{\sqrt{5n^2+7n-9}}{\sqrt[3]{7n^4+5}}

Problem set

For each of the following problems, give, with a justification, whether the given series converges or diverges.

  1. \sum\limits_{n=1}^{\infty}\sin\left(\frac{1}{n}\right)

  2. \sum\limits_{n=1}^{\infty}\sin\left(\frac{1}{n^2}\right)

  3. \sum\limits_{n=1}^{\infty}\tan\left(\frac{1}{\sqrt{n}}\right)

Series with positive and negative terms

Exercises

Problem set

For each of the following problems, answer with justification, whether the given series is absolutely convergent or conditionally convergent or divergent.

  1. \sum\limits_{n=1}^{\infty} \frac{(-1)^n}{n^2}

  2. \sum\limits_{n=1}^{\infty} \frac{(-1)^n}{n}

  3. \sum\limits_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}

  4. \sum\limits_{n=1}^{\infty} \frac{(-1)^n}{e^{1/n}}

Problem set

For each of the following problems, answer with justification, whether the given series is absolutely convergent or conditionally convergent or divergent.

  1. \sum\limits_{n=1}^{\infty} \frac{\cos n}{n^2}

  2. \sum\limits_{n=1}^{\infty} \frac{(-1)^{\left\lfloor\frac{n}{3}\right\rfloor}}{n^2}. Note that \lfloor x \rfloor is the floor of (greatest integer less than or equal to) x.

  3. \sum\limits_{n=1}^{\infty} \frac{n\sin n}{n^3+5}

  4. \sum\limits_{n=1}^{\infty} \frac{n\cos n}{\sqrt{n^5-2}}

Ratio and root tests

Exercises

Problem set

For each of the following problems, give, with a justification, whether the given series converges or diverges.

  1. \sum\limits_{n=1}^{\infty} \frac{n^{101}}{1.01^n}

  2. \sum\limits_{n=1}^{\infty} \frac{3^n}{n!}

  3. \sum\limits_{n=1}^{\infty} \frac{n!n!}{(2n)!}

  4. \sum\limits_{n=1}^{\infty} \frac{(-1)^n (2n)!}{2^nn!n!}

Problem set

For each of the following problems, give, with a justification, whether the given series converges or diverges.

  1. \sum\limits_{n=1}^{\infty} \frac{9^n}{n^n}

  2. \sum\limits_{n=1}^{\infty} \left(\frac{2n+1}{3n+5}\right)^n

Power series

Exercises

Problem set

For each of the following problems, give, for what values of x, the given power series converges.

  1. \sum\limits_{n=1}^{\infty} x^n

  2. \sum\limits_{n=1}^{\infty} x^{2n}

  3. \sum\limits_{n=1}^{\infty} (-x)^n

  4. \sum\limits_{n=1}^{\infty} \frac{(-1)^nx^n}{n}

  5. \sum\limits_{n=1}^{\infty} \frac{n+1}{n+2}x^n

Problem set

For each of the following problems, give the interval of convergence for the given power series, and give the function that the power series converges to.

  1. \sum\limits_{n=0}^{\infty} (3x)^n

  2. \sum\limits_{n=0}^{\infty} (5x)^{2n}

  3. \sum\limits_{n=0}^{\infty} (-1)^{n+1}x^{2n}

  4. \sum\limits_{n=0}^{\infty} (n+1)x^n

  5. \sum\limits_{n=0}^{\infty} (-1)^n(2n+2)x^{2n+1}

Problem set

For each of the following problems, give the interval of convergence for the given power series, and give the function that the power series converges to.

  1. \sum\limits_{n=0}^{\infty} (x-3)^n

  2. \sum\limits_{n=0}^{\infty} n!(x+5)^n

Maclaurin series and Taylor series

Exercises

Problem set

In each of the following cases, estimate the maximum error of the approximation, as suggested by the Taylor’s theorem.

  1. Approximation of e^5 by the order 3 Taylor polynomial of e^x about x=0

  2. Approximation of \tan\left(\frac{\pi}{6}\right) by the order 2 Taylor polynomial of \tan x about x=0

  3. Approximation of \ln(1.8) by the order 3 Taylor polynomial of \ln(1+x) about x=0

Problem set

In each of the following problems, a function f(x) and a value of x at which the function value $f(x)$ is approximately desired is given. Also, the maximum approximation error is given. Determine the smallest order n of Taylor polynomial about x=0 that needs to be used to guarantee that the approximation error is within the maximum bound.

  1. f(x) = \sin x at x = 0.5 and Maximum approximation error: 0.001

Problem set

In each of the following problems, a function and a Taylor polynomial for the function are given. Also, the maximum allowed error for approximating the function by the Taylor polynomial is given. Determine for what values of x, the Taylor polynomial can approximate the function while satisfying the error requirement.

  1. Function: e^x, Taylor polynomial: 1 + \frac{x}{1!} + \frac{x^2}{2!}, Maximum allowed error: 0.01

  2. Function: \ln(1+x), Taylor polynomial: x - \frac{x^2}{2} + \frac{x^3}{3}, Maximum allowed error: 0.01

  3. Function: \cos x, Taylor polynomial: 1 - \frac{x^2}{2!} + \frac{x^4}{4!}, Maximum allowed error: 0.0001

Problem set

For each of the following problems, give the Maclaurin series for the given function.

  1. e^x

  2. \sin x

  3. \cos x

  4. \ln (x+1)