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Exercises relating to the direct and limit comparison tests for series.

Suppose you wished to use the Direct Comparison Test to establish convergence of the series Which of the following options below would be valid series for comparison? Select all that apply.
$\displaystyle \sum _{n=2}^\infty \frac {1}{n^5}$ $\displaystyle \sum _{n=2}^\infty \frac {1}{n^4}$ $\displaystyle \sum _{n=2}^\infty \frac {1}{n^3}$ $\displaystyle \sum _{n=2}^\infty \frac {1}{n^2}$ $\displaystyle \sum _{n=2}^\infty \frac {1}{n}$

Suppose you wished to use the Direct Comparison Test to establish convergence of the series Which of the following options below would be valid series for comparison? Select all that apply.
$\displaystyle \sum _{n=2}^\infty \frac {1}{n^4}$ $\displaystyle \sum _{n=2}^\infty \frac {1}{n^3}$ $\displaystyle \sum _{n=2}^\infty \frac {1}{n^2}$

Using only the comparison functions listed here: which of the following series can be proved to either diverge or converge using the Direct Comparison Test? (Select all that apply.)
$\displaystyle \sum _{n=1}^\infty \frac {1}{n^{3/2}+n^{1/2}}$ $\displaystyle \sum _{n=1}^\infty \frac {1}{n^2 + n^{1/2}}$ $\displaystyle \sum _{n=1}^\infty \frac {1}{\ln (n+1)}$ $\displaystyle \sum _{n=2}^\infty \frac {1}{n^2-1}$ $\displaystyle \sum _{n=1}^\infty \frac {1}{e^n + 1}$ $\displaystyle \sum _{n=3}{\infty } \frac {1}{2n+3}$

Suppose you wished to use the Limit Comparison Test to establish convergence of the series Which of the following options below would be valid series for comparison? Select all that apply.
$\displaystyle \sum _{n=2}^\infty \frac {1}{n^5}$ $\displaystyle \sum _{n=2}^\infty \frac {1}{n^4}$ $\displaystyle \sum _{n=2}^\infty \frac {1}{n^3}$ $\displaystyle \sum _{n=2}^\infty \frac {1}{n^2}$ $\displaystyle \sum _{n=2}^\infty \frac {1}{n}$

Suppose you wished to use the Limit Comparison Test to establish convergence of the series Which of the following options below would be valid series for comparison? Select all that apply.
$\displaystyle \sum _{n=2}^\infty \frac {1}{n^4}$ $\displaystyle \sum _{n=2}^\infty \frac {1}{n^3}$ $\displaystyle \sum _{n=2}^\infty \frac {1}{n^2}$
Using only the comparison functions listed here: , which of the following series can be proved to either diverge or converge using the Limit Comparison Test? (Select all that apply.)
$\displaystyle \sum _{n=1}^\infty \frac {1}{n^{3/2}+n^{1/2}}$ $\displaystyle \sum _{n=1}^\infty \frac {1}{n^2 + n^{1/2}}$ $\displaystyle \sum _{n=1}^\infty \frac {1}{\ln (n+1)}$ $\displaystyle \sum _{n=2}^\infty \frac {1}{n^2-1}$ $\displaystyle \sum _{n=1}^\infty \frac {1}{e^n + 1}$ $\displaystyle \sum _{n=3}{\infty } \frac {1}{2n+3}$