Chapter 11: Sequences and SeriesContent check — Calculus (Guichard), Chapter 11: Sequences and Series

Compute ${\displaystyle \underset{x\to \infty }{\lim }{x}^{1/x}}$⁠.

Determine whether ${\displaystyle {\{\sqrt{n+47}-\sqrt{n}\}}_{n=0}^{\infty }}$ converges or diverges. If it converges, compute the limit.

Determine whether ${\displaystyle {\left\{\frac{n^2+1}{{(n+1)}^2}\right\}}_{n=0}^{\infty }}$ converges or diverges. If it converges, compute the limit.

Determine whether ${\displaystyle {\left\{\frac{n+47}{\sqrt{n^2+3n}}\right\}}_{n=1}^{\infty }}$ converges or diverges. If it converges, compute the limit.

Determine whether ${\displaystyle {\left\{\frac{2^n}{n!}\right\}}_{n=0}^{\infty }}$ converges or diverges.

Explain why ${\displaystyle \sum _{n=1}^{\infty }\frac{n^2}{2n^2+1}}$ diverges.

Explain why ${\displaystyle \sum _{n=1}^{\infty }\frac{5}{2^{1/n}+14}}$ diverges.

Explain why ${\displaystyle \sum _{n=1}^{\infty }\frac{3}{n}}$ diverges.

Compute ${\displaystyle \sum _{n=0}^{\infty }\frac{4}{{(-3)}^n}-\frac{3}{3^n}}$⁠.

Compute ${\displaystyle \sum _{n=0}^{\infty }\frac{3}{2^n}+\frac{4}{5^n}}$⁠.

Compute ${\displaystyle \sum _{n=0}^{\infty }\frac{4^{n+1}}{5^n}}$⁠.

Compute ${\displaystyle \sum _{n=0}^{\infty }\frac{3^{n+1}}{7^{n+1}}}$⁠.

Compute ${\displaystyle \sum _{n=1}^{\infty }{\left(\frac{3}{5}\right)}^n}$⁠.

Compute ${\displaystyle \sum _{n=1}^{\infty }\frac{3^n}{5^{n+1}}}$⁠.

Determine whether each series converges or diverges.

${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^{\pi /4}}}$

Determine whether each series converges or diverges.

${\displaystyle \sum _{n=1}^{\infty }\frac{n}{n^2+1}}$

Determine whether each series converges or diverges.

${\displaystyle \sum _{n=1}^{\infty }\frac{\ln n}{n^2}}$

Determine whether each series converges or diverges.

${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2+1}}$

Determine whether each series converges or diverges.

${\displaystyle \sum _{n=1}^{\infty }\frac{1}{e^n}}$

Determine whether each series converges or diverges.

${\displaystyle \sum _{n=1}^{\infty }\frac{n}{e^n}}$

Determine whether each series converges or diverges.

${\displaystyle \sum _{n=2}^{\infty }\frac{1}{n\ln n}}$

Determine whether each series converges or diverges.

${\displaystyle \sum _{n=2}^{\infty }\frac{1}{n{(\ln n)}^2}}$

Determine whether each series converges or diverges.

Find an $N$ so that ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^4}}$ is between ${\displaystyle \sum _{n=1}^N\frac{1}{n^4}}$ and ${\displaystyle \sum _{n=1}^N\frac{1}{n^4}+0.005}$⁠.

Determine whether each series converges or diverges.

Find an $N$ so that ${\displaystyle \sum _{n=0}^{\infty }\frac{1}{e^n}}$ is between ${\displaystyle \sum _{n=0}^N\frac{1}{e^n}}$ and ${\displaystyle \sum _{n=0}^N\frac{1}{e^n}+{10}^{-4}}$⁠.

Determine whether each series converges or diverges.

Find an $N$ so that ${\displaystyle \sum _{n=1}^{\infty }\frac{\ln n}{n^2}}$ is between ${\displaystyle \sum _{n=1}^N\frac{\ln n}{n^2}}$ and ${\displaystyle \sum _{n=1}^N\frac{\ln n}{n^2}+0.005}$⁠.

Determine whether each series converges or diverges.

Find an $N$ so that ${\displaystyle \sum _{n=2}^{\infty }\frac{1}{n{(\ln n)}^2}}$ is between ${\displaystyle \sum _{n=2}^N\frac{1}{n{(\ln n)}^2}}$ and ${\displaystyle \sum _{n=2}^N\frac{1}{n{(\ln n)}^2}+0.005}$⁠.

Determine whether the following series converge or diverge.

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

Determine whether the following series converge or diverge.

${\displaystyle \sum _{n=4}^{\infty }\frac{{(-1)}^{n-1}}{\sqrt{n-3}}}$

Determine whether the following series converge or diverge.

${\displaystyle \sum _{n=1}^{\infty }{(-1)}^{n-1}\frac{n}{3n-2}}$

Determine whether the following series converge or diverge.

${\displaystyle \sum _{n=1}^{\infty }{(-1)}^{n-1}\frac{\ln n}{n}}$

Determine whether the following series converge or diverge.

Approximate ${\displaystyle \sum _{n=1}^{\infty }{(-1)}^{n-1}\frac{1}{n^3}}$ to two decimal places.

Determine whether the following series converge or diverge.

Approximate ${\displaystyle \sum _{n=1}^{\infty }{(-1)}^{n-1}\frac{1}{n^4}}$ to two decimal places.

Determine whether the series converge or diverge.

${\displaystyle \sum _{n=1}^{\infty }\frac{1}{2n^2+3n+5}}$

Determine whether the series converge or diverge.

${\displaystyle \sum _{n=2}^{\infty }\frac{1}{2n^2+3n-5}}$

Determine whether the series converge or diverge.

${\displaystyle \sum _{n=1}^{\infty }\frac{1}{2n^2-3n-5}}$

Determine whether the series converge or diverge.

${\displaystyle \sum _{n=1}^{\infty }\frac{3n+4}{2n^2+3n+5}}$

Determine whether the series converge or diverge.

${\displaystyle \sum _{n=1}^{\infty }\frac{3n^2+4}{2n^2+3n+5}}$

Determine whether the series converge or diverge.

${\displaystyle \sum _{n=1}^{\infty }\frac{\ln n}{n}}$

Determine whether the series converge or diverge.

${\displaystyle \sum _{n=1}^{\infty }\frac{\ln n}{n^3}}$

Determine whether the series converge or diverge.

${\displaystyle \sum _{n=2}^{\infty }\frac{1}{\ln n}}$

Determine whether the series converge or diverge.

${\displaystyle \sum _{n=1}^{\infty }\frac{3^n}{2^n+5^n}}$

Determine whether the series converge or diverge.

${\displaystyle \sum _{n=1}^{\infty }\frac{3^n}{2^n+3^n}}$

Determine whether each series converges absolutely, converges conditionally, or diverges.

${\displaystyle \sum _{n=1}^{\infty }{(-1)}^{n-1}\frac{1}{2n^2+3n+5}}$

Determine whether each series converges absolutely, converges conditionally, or diverges.

${\displaystyle \sum _{n=1}^{\infty }{(-1)}^{n-1}\frac{3n^2+4}{2n^2+3n+5}}$

Determine whether each series converges absolutely, converges conditionally, or diverges.

${\displaystyle \sum _{n=1}^{\infty }{(-1)}^{n-1}\frac{\ln n}{n}}$

Determine whether each series converges absolutely, converges conditionally, or diverges.

${\displaystyle \sum _{n=1}^{\infty }{(-1)}^{n-1}\frac{\ln n}{n^3}}$

Determine whether each series converges absolutely, converges conditionally, or diverges.

${\displaystyle \sum _{n=2}^{\infty }{(-1)}^n\frac{1}{\ln n}}$

Determine whether each series converges absolutely, converges conditionally, or diverges.

${\displaystyle \sum _{n=0}^{\infty }{(-1)}^n\frac{3^n}{2^n+5^n}}$

Determine whether each series converges absolutely, converges conditionally, or diverges.

${\displaystyle \sum _{n=0}^{\infty }{(-1)}^n\frac{3^n}{2^n+3^n}}$

Determine whether each series converges absolutely, converges conditionally, or diverges.

${\displaystyle \sum _{n=1}^{\infty }{(-1)}^{n-1}\frac{\arctan n}{n}}$

Determine whether the series converge.

${\displaystyle \sum _{n=0}^{\infty }{(-1)}^n\frac{3^n}{5^n}}$

Determine whether the series converge.

${\displaystyle \sum _{n=1}^{\infty }\frac{n!}{n^n}}$

Determine whether the series converge.

${\displaystyle \sum _{n=1}^{\infty }\frac{n^5}{n^n}}$

Determine whether the series converge.

${\displaystyle \sum _{n=1}^{\infty }\frac{{(n!)}^2}{n^n}}$

Find the radius and interval of convergence for each series. In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence.

${\displaystyle \sum _{n=0}^{\infty }n{x}^n}$

Find the radius and interval of convergence for each series. In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence.

${\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n!}}$

Find the radius and interval of convergence for each series. In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence.

${\displaystyle \sum _{n=1}^{\infty }\frac{n!}{n^n}{x}^n}$

Find the radius and interval of convergence for each series. In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence.

${\displaystyle \sum _{n=1}^{\infty }\frac{n!}{n^n}{(x-2)}^n}$

Find the radius and interval of convergence for each series. In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence.

${\displaystyle \sum _{n=1}^{\infty }\frac{{(n!)}^2}{n^n}{(x-2)}^n}$

Find the radius and interval of convergence for each series. In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence.

${\displaystyle \sum _{n=1}^{\infty }\frac{{(x+5)}^n}{n(n+1)}}$

Find a series representation for $\ln 2$⁠.

Find a power series representation for ${\displaystyle 1/{(1-x)}^2}$⁠.

Find a power series representation for ${\displaystyle 2/{(1-x)}^3}$⁠.

Find a power series representation for ${\displaystyle 1/{(1-x)}^3}$⁠. What is the radius of convergence?

Find a power series representation for ${\displaystyle \int \ln (1-x)dx}$⁠.

For each function, find the Maclaurin series or Taylor series centered at $a$⁠, and the radius of convergence.

$\cos x$

For each function, find the Maclaurin series or Taylor series centered at $a$⁠, and the radius of convergence.

${\displaystyle e^x}$

For each function, find the Maclaurin series or Taylor series centered at $a$⁠, and the radius of convergence.

$1/x$⁠, $a=5$

For each function, find the Maclaurin series or Taylor series centered at $a$⁠, and the radius of convergence.

$\ln x$⁠, $a=1$

For each function, find the Maclaurin series or Taylor series centered at $a$⁠, and the radius of convergence.

$\ln x$⁠, $a=2$

For each function, find the Maclaurin series or Taylor series centered at $a$⁠, and the radius of convergence.

${\displaystyle 1/x^2}$⁠, $a=1$

For each function, find the Maclaurin series or Taylor series centered at $a$⁠, and the radius of convergence.

${\displaystyle 1/\sqrt{1-x}}$

For each function, find the Maclaurin series or Taylor series centered at $a$⁠, and the radius of convergence.

Find the first four terms of the Maclaurin series for $\tan x$ (up to and including the ${\displaystyle x^3}$ term).

For each function, find the Maclaurin series or Taylor series centered at $a$⁠, and the radius of convergence.

Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for ${\displaystyle x\cos (x^2)}$⁠.

For each function, find the Maclaurin series or Taylor series centered at $a$⁠, and the radius of convergence.

Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for ${\displaystyle x{e}^{-x}}$⁠.

Find a polynomial approximation for $\cos x$ on $[0,\pi ]$⁠, accurate to ${\displaystyle \pm {10}^{-3}}$

How many terms of the series for $\ln x$ centered at 1 are required so that the guaranteed error on $[1/\mathrm{2,3}/2]$ is at most ${\displaystyle {10}^{-3}}$⁠? What if the interval is instead $[\mathrm{1,3}/2]$⁠?

Find the first three nonzero terms in the Taylor series for $\tan x$ on $[-\pi /4,\pi /4]$⁠, and compute the guaranteed error term as given by Taylor's theorem. (You may want to use Sage or a similar aid.)

Determine whether the series converges.

${\displaystyle \sum _{n=0}^{\infty }\frac{n}{n^2+4}}$

Determine whether the series converges.

${\displaystyle \frac{1}{1\cdot 2}+\frac{1}{3\cdot 4}+\frac{1}{5\cdot 6}+\frac{1}{7\cdot 8}+\cdots }$

Determine whether the series converges.

${\displaystyle \sum _{n=0}^{\infty }\frac{n}{{(n^2+4)}^2}}$

Determine whether the series converges.

${\displaystyle \sum _{n=0}^{\infty }\frac{n!}{8^n}}$

Determine whether the series converges.

${\displaystyle 1-\frac{3}{4}+\frac{5}{8}-\frac{7}{12}+\frac{9}{16}+\cdots }$

Determine whether the series converges.

${\displaystyle \sum _{n=0}^{\infty }\frac{1}{\sqrt{n^2+4}}}$

Determine whether the series converges.

${\displaystyle \sum _{n=0}^{\infty }\frac{{\sin }^3(n)}{n^2}}$

Determine whether the series converges.

${\displaystyle \sum _{n=0}^{\infty }\frac{n}{e^n}}$

Determine whether the series converges.

${\displaystyle \sum _{n=0}^{\infty }\frac{n!}{1\cdot 3\cdot 5\cdots (2n-1)}}$

Determine whether the series converges.

${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n\sqrt{n}}}$

Determine whether the series converges.

${\displaystyle \frac{1}{2\cdot 3\cdot 4}+}$ ${\displaystyle \frac{2}{3\cdot 4\cdot 5}+}$ ${\displaystyle \frac{3}{4\cdot 5\cdot 6}+}$ ${\displaystyle \frac{4}{5\cdot 6\cdot 7}+}$ ${\displaystyle \cdots }$

Determine whether the series converges.

${\displaystyle \sum _{n=1}^{\infty }\frac{1\cdot 3\cdot 5\cdots (2n-1)}{(2n)!}}$

Determine whether the series converges.

${\displaystyle \sum _{n=0}^{\infty }\frac{6^n}{n!}}$

Determine whether the series converges.

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

Determine whether the series converges.

${\displaystyle \sum _{n=1}^{\infty }\frac{2^n{3}^{n-1}}{n!}}$

Determine whether the series converges.

${\displaystyle 1+}$ ${\displaystyle \frac{5^2}{2^2}+}$ ${\displaystyle \frac{5^4}{{(2\cdot 4)}^2}+}$ ${\displaystyle \frac{5^6}{{(2\cdot 4\cdot 6)}^2}+}$ ${\displaystyle \frac{5^8}{{(2\cdot 4\cdot 6\cdot 8)}^2}+}$ ${\displaystyle \cdots }$

Determine whether the series converges.

${\displaystyle \sum _{n=1}^{\infty }\sin (1/n)}$

Find the interval and radius of convergence; you need not check the endpoints of the intervals.

${\displaystyle \sum _{n=0}^{\infty }\frac{2^n}{n!}{x}^n}$

Find the interval and radius of convergence; you need not check the endpoints of the intervals.

${\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{1+3^n}}$

Find the interval and radius of convergence; you need not check the endpoints of the intervals.

${\displaystyle \sum _{n=1}^{\infty }\frac{x^n}{n{3}^n}}$

Find the interval and radius of convergence; you need not check the endpoints of the intervals.

${\displaystyle x+}$ ${\displaystyle \frac{1}{2}\frac{x^3}{3}+}$ ${\displaystyle \frac{1\cdot 3}{2\cdot 4}\frac{x^5}{5}+}$ ${\displaystyle \frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{x^7}{7}+}$ ${\displaystyle \cdots }$

Find the interval and radius of convergence; you need not check the endpoints of the intervals.

${\displaystyle \sum _{n=1}^{\infty }\frac{n!}{n^2}{x}^n}$

Find the interval and radius of convergence; you need not check the endpoints of the intervals.

${\displaystyle \sum _{n=1}^{\infty }\frac{{(-1)}^n}{n^2{3}^n}{x}^{2n}}$

Find the interval and radius of convergence; you need not check the endpoints of the intervals.

${\displaystyle \sum _{n=0}^{\infty }\frac{{(x-1)}^n}{n!}}$

Find a series for each function, using the formula for Maclaurin series and algebraic manipulation as appropriate.

${\displaystyle 2^x}$

Find a series for each function, using the formula for Maclaurin series and algebraic manipulation as appropriate.

${\displaystyle \ln (1+x)}$

Find a series for each function, using the formula for Maclaurin series and algebraic manipulation as appropriate.

${\displaystyle \ln \left(\frac{1+x}{1-x}\right)}$

Find a series for each function, using the formula for Maclaurin series and algebraic manipulation as appropriate.

${\displaystyle \sqrt{1+x}}$

Find a series for each function, using the formula for Maclaurin series and algebraic manipulation as appropriate.

${\displaystyle \frac{1}{1+x^2}}$

Find a series for each function, using the formula for Maclaurin series and algebraic manipulation as appropriate.

${\displaystyle \arctan (x)}$

Find a series for each function, using the formula for Maclaurin series and algebraic manipulation as appropriate.

Use the answer to the previous problem to discover a series for a well-known mathematical constant.