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

True or false: If $A$ is invertible, then $A$ is diagonalizable.

Problem 5

True or false: If $A$ and $B$ are invertible $n\times n$ matrices, then so is $A+B.$

Problem 6

True or false: If $A$ is a $3\times 5$ matrix and ${\text{dim Nul }}A=2,$ then $A{\vec {x}}={\vec {b}}$ is consistent for all ${\vec {b}}$ in $\mathbb {R} ^{3}.$

Problem 7

True or false: Let $C=AB$ for $4\times 4$ matrices $A$ and $B.$ If $C$ is invertible, then $A$ is invertible.

Problem 8

True or false: Let $W$ be a subspace of $\mathbb {R} ^{4}$ and ${\vec {v}}$ be a vector in $\mathbb {R} ^{4}.$ If ${\vec {v}}\in W$ and ${\vec {v}}\in W^{\perp },$ then ${\vec {v}}={\vec {0}}.$