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 

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

 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 

$\sum _{n=1}^{\infty }{\frac {10^{n}}{(n+1)4^{2n+1}}}$

 Problem 7 

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

 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  $C$  are  $3\times 3$  matrices such that  $AB=AC,$  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.}