Series Problems

These questions are meant to be practice problems for series.

Determine whether the series converge or diverge.

Click on the boxed problem numbers to go to a solution.

Problem 1

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty \frac{2+3^n}{4^n}}

Problem 2

\sum _{n=1}^{\infty }\ln {\Bigg (}{\frac {n^{2}+1}{2n^{2}+1}}{\Bigg )}

Problem 3

\sum _{n=1}^{\infty }{\frac {n}{n^{4}+1}}

Problem 4

\sum _{n=1}^{\infty }{\frac {n^{2}-1}{3n^{4}+1}}

Problem 5

\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}n^{2}}{10^{n}}}

Problem 6

Failed to parse (syntax error): {\displaystyle \sum_{n=1}^\infty \frac{10^n}{(n+1)4^{2n+1}} == [[Series Problems,_Problem_7|<span class="biglink"><span style="font-size:80%"> Problem 7 </span>]] == ::<span class="exam"><math>\sum_{n=1}^\infty \frac{(n^2+1)^{2n}}{(2n^2+1)^n}}

Problem 8

True or false: Let $W$ be a subspace of $\mathbb {R} ^{4}$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {v}}} be a vector in $\mathbb {R} ^{4}.$ If ${\vec {v}}\in W$ and ${\vec {v}}\in W^{\perp },$ then ${\vec {v}}={\vec {0}}.$

Problem 9

True or false: If $A$ is an invertible $3\times 3$ matrix, and $B$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\times 3} matrices such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AB=AC,} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B=C.}