Consider the sequence \( \displaystyle a_n=\frac{3n^2-6n+7}{4n^2+8n+5} \).
Determine whether the series \( \displaystyle \sum_{n=5}^{\infty}\frac{n^{2}-6n^{5}+5n^{6} }{ n^{4}+n^{8}-3} \) is convergent or divergent.
Assume that \(\overrightarrow u=\langle -1, 9\rangle\), \(\overrightarrow v= \langle -2, 9\rangle\), and \(\overrightarrow w= \langle 5, -2\rangle \). Let \( \overrightarrow z \) be the vector defined as \[ \overrightarrow z=4\overrightarrow u+ 2\overrightarrow v -3\overrightarrow w.\] Determine \( \displaystyle \overrightarrow z\cdot\langle1, 1\rangle \).
Assume that \( \overrightarrow u=\langle 4, 0,6\rangle \) and \( \overrightarrow v=\langle 0,1,4\rangle \).
Determine the value of the variable \( x \) for which the vectors \( \overrightarrow u=\langle 8,36\rangle \) and \( \overrightarrow v=\langle 72,x\rangle \) are orthogonal.
Determine the interval of convergence of the power series \( \displaystyle \sum_{n=1}^{\infty}\frac1{(n+8)^{\frac{4}{5}}}\cdot (5-7x)^n \).
For each of the series determine whether it is absolutely convergent, conditionally convergent, or divergent. Provide mathematically rigorous justifications for your answers.
Assume that \( f \) is a function whose corresponding Taylor polynomial of degree \( 5 \) around \( a=2 \) is given by: \[ T_5(x)=1+2(x-2)+3(x-2)^2+4(x-2)^3-5(x-2)^4+5(x-2)^5.\] Determine \( f^{\prime\prime\prime}(2) \).
For each of the following statements determine whether it is true or false. If the statement is true, prove it. If it is false, provide a counter-example.