Question

1. (10 points) Calculate the improper integral if it converges, or show that it diverges.\\ $\int_1^\infty \frac{7}{(3x+1)^4} dx$\\ 2. (5 points) Consider the sequence defined by $a_0 = 0, a_1 = 1$ and\\ $a_{n+2} = 3a_n + 4a_{n+1}$\\ for $n \ge 0$.\\ (a) Is this sequence defined implicitly or explicitly? Explain your answer.\\ (b) Write down all of the terms in the sequence that are below 300.\\ 3. (10 points) Show all of your work in this problem. Consider the infinite series\\ $\sum_{k=2}^\infty \frac{k-1}{2k+1}$\\ (a) Write down the first three terms of this series.\\ (b) Find the first three terms in the sequence of partial sums of this series.\\ In parts (a) and (b), write your answers as reduced fractions.

          1. (10 points) Calculate the improper integral if it converges, or show that it diverges.\\
$\int_1^\infty \frac{7}{(3x+1)^4} dx$\\
2. (5 points) Consider the sequence defined by $a_0 = 0, a_1 = 1$ and\\
$a_{n+2} = 3a_n + 4a_{n+1}$\\
for $n \ge 0$.\\
(a) Is this sequence defined implicitly or explicitly? Explain your answer.\\
(b) Write down all of the terms in the sequence that are below 300.\\
3. (10 points) Show all of your work in this problem. Consider the infinite series\\
$\sum_{k=2}^\infty \frac{k-1}{2k+1}$\\
(a) Write down the first three terms of this series.\\
(b) Find the first three terms in the sequence of partial sums of this series.\\
In parts (a) and (b), write your answers as reduced fractions.

$1. (10 points) Calculate the improper integral if it converges, or show that it diverges. ∫1^∞(7)/((3x+1)^4) dx 2. (5 points) Consider the sequence defined by a0 = 0, a1 = 1 and an+2 = 3an + 4an+1 for n ≥ 0. (a) Is this sequence defined implicitly or explicitly? Explain your answer. (b) Write down all of the terms in the sequence that are below 300. 3. (10 points) Show all of your work in this problem. Consider the infinite series ∑k=2^∞(k-1)/(2k+1) (a) Write down the first three terms of this series. (b) Find the first three terms in the sequence of partial sums of this series. In parts (a) and (b), write your answers as reduced fractions.$

Added by Consuelo I.

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