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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.