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MTH 3030: Midterm 1 Practice 1

Problem 1

Consider the sequence \( \displaystyle a_n=\frac{3n^2-6n+7}{4n^2+8n+5} \).

  • (a) Determine the limit \( \displaystyle \lim_{n\to\infty} a_n \).

  • (b) State the divergence test.

  • (c) Determine whether the series \( \displaystyle \sum_{n=1}^{\infty}a_n \) is convergent or divergent.

Problem 2

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.

Problem 3

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 \( \displaystyle \overrightarrow z\cdot\langle1, 1\rangle \).

Problem 4

Assume that \( \overrightarrow u=\langle 4, 0,6\rangle \) and \( \overrightarrow v=\langle 0,1,4\rangle \).

  • (a) Find the cross product \( \overrightarrow u\times\overrightarrow v \).

  • (b) Determine a unit vector that is orthogonal to both \( \overrightarrow u \) and \( \overrightarrow v \).

Problem 5

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.

Problem 6

Determine the interval of convergence of the power series \( \displaystyle \sum_{n=1}^{\infty}\frac1{(n+8)^{\frac{4}{5}}}\cdot (5-7x)^n \).

Problem 7

For each of the series determine whether it is absolutely convergent, conditionally convergent, or divergent. Provide mathematically rigorous justifications for your answers.

  • (a) \( \displaystyle \sum_{n=7}^{\infty}\frac{(-1)^n}{8+\ln n} \);

  • (b) \( \displaystyle \sum_{n=7}^{\infty}\frac{(-1)^n}{9+(-2)^n} \);

  • (c) \( \displaystyle \sum_{n=1}^{\infty}\frac{n^{6}}{8^n} \);

  • (d) \( \displaystyle \sum_{n=3}^{\infty}\frac1{\left(\ln n\right)^{\sqrt{\ln n}}} \).

Problem 8

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 \( f^{\prime\prime\prime}(2) \).

Problem 9

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.

  • (a) If \((a_n)\) and \((b_n)\) are sequences such that \(a_n\leq b_n\) for all \(n\) and \(\sum_{n=1}^{\infty}b_n\) is convergent, then \(\displaystyle\sum_{n=1}^{\infty}a_n\) is convergent as well.

  • (b) If \((a_n)\) and \((b_n)\) are sequences with positive terms such that \(a_n\leq b_n\) for all \(n\) and \(\sum_{n=1}^{\infty}b_n\) is convergent, then \(\displaystyle\sum_{n=1}^{\infty}a_n\) is convergent as well.

  • (c) If \((a_n)\) and \((b_n)\) are sequences with positive terms such that \(\displaystyle \sum_{n=1}^{\infty}a_n\) and \(\displaystyle \sum_{n=1}^{\infty}b_n\) are convergent, then the series \(\displaystyle \sum_{n=1}^{\infty}a_nb_n\) is convergent as well.

  • (d) If \((a_n)\) and \((b_n)\) are sequences such that \(\displaystyle \sum_{n=1}^{\infty}a_n\) and \(\displaystyle \sum_{n=1}^{\infty}b_n\) are convergent, then the series \(\displaystyle \sum_{n=1}^{\infty}a_nb_n\) is convergent as well.