Miscellaneous

Evaluate the following sums

  1. $\sum_{{n=1}}^{{\infty }}$${\frac{{1}}{{n^{2}}}}$

  2. $\sum_{{n=1}}^{{100}}$n4

  3. Find the cube roots of the number 1 + i.

  4. Compute the determinant of the matrix $\left(\vphantom{
\begin{array}{lll}
1 & 2 & 3 \\
2 & 4 & 5 \\
3 & 5 & 6
\end{array}
}\right.$$\begin{array}{lll}
1 & 2 & 3 \\
2 & 4 & 5 \\
3 & 5 & 6
\end{array}$$\left.\vphantom{
\begin{array}{lll}
1 & 2 & 3 \\
2 & 4 & 5 \\
3 & 5 & 6
\end{array}
}\right)$.

  5. Solve the equation $\left(\vphantom{
\begin{array}{ll}
2 & 3 \\
4 & 5
\end{array}
}\right.$$\begin{array}{ll}
2 & 3 \\
4 & 5
\end{array}$$\left.\vphantom{
\begin{array}{ll}
2 & 3 \\
4 & 5
\end{array}
}\right)$X = $\left(\vphantom{
\begin{array}{ll}
3 & 7 \\
8 & 9
\end{array}
}\right.$$\begin{array}{ll}
3 & 7 \\
8 & 9
\end{array}$$\left.\vphantom{
\begin{array}{ll}
3 & 7 \\
8 & 9
\end{array}
}\right)$.

  6. Find a basis for the nullspace of $\left(\vphantom{
\begin{array}{ccc}
3 & 1 & 0 \\
3 & 1 & 6 \\
6 & 2 & 0
\end{array}
}\right.$$\begin{array}{ccc}
3 & 1 & 0 \\
3 & 1 & 6 \\
6 & 2 & 0
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
3 & 1 & 0 \\
3 & 1 & 6 \\
6 & 2 & 0
\end{array}
}\right)$.

  7. Solve the system of equations
    x + 2y + 3z = 1
    2x + 4y + 5z = 1
    3x + 5x + 6z = 1
    .

  8. Use Newton's method to find a solution to the equation sin x - x3 = 0 for x > 0.