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Exercises relating to various topics we have studied.

Determine whether the series below converges absolutely, conditionally, or diverges.
Absolute Conditional Diverge
Determine whether the series below converges absolutely, conditionally, or diverges.
Absolute Conditional Diverge
Take absolute values and try direct comparison test.
Try direct comparison to $e^{-n}$.
Determine whether the series below converges absolutely, conditionally, or diverges.
Absolute Conditional Diverge
What is the dominant term in the denominator as $n \rightarrow \infty$?
The dominant term in the denominator is $n$.
This means that the terms do not go to zero.
Determine whether the series below converges absolutely, conditionally, or diverges.
Absolute Conditional Diverge
What is the dominant term in the denominator as $n \rightarrow \infty$?
The dominant term in the denominator is $n^2$.
Determine whether the series below converges absolutely, conditionally, or diverges.
Absolute Conditional Diverge
Determine whether the series below converges absolutely, conditionally, or diverges.
Absolute Conditional Diverge
Which term dominates as $n \rightarrow \infty$: $\sqrt {n}$ or $\ln (n+1)$?
Answer: $\sqrt {n}$ dominates $\ln (n+1)$ as $n \rightarrow \infty$.
Determine whether the series below converges absolutely, conditionally, or diverges.
Absolute Conditional Diverge
To show that convergence is not absolute, try limit comparison.
The comparison series can be taken to be $1/\sqrt {n}$ in this case.
Determine whether the series below converges absolutely, conditionally, or diverges.
Absolute Conditional Diverge
Try limit comparison with the harmonic series.
Determine whether the series below converges absolutely, conditionally, or diverges.
Absolute Conditional Diverge
With absolute values, compare to a harmonic series.
How do we know that the alternating series test applies?

### Sample Exam Questions

Determine whether the following series converge or diverge.

I & II converge; III & IV diverge I & III converge; II & IV diverge I & IV converge; II & III diverge II & III converge; I & IV diverge II & IV converge; I & III diverge III & IV converge; I & II diverge

Determine whether the following series are convergent or divergent. Justify your answers.

I & II divergent I convergent, II divergent I divergent, II convergent I & II convergent

Determine whether the following series are convergent or divergent. Justify your answers.

I & II divergent I convergent, II divergent I divergent, II convergent I & II convergent

Determine which of the following series are convergent. For full credit, be sure to explain your reasoning and specify which tests were used.

only I only I and II only I and III only II only II and III only III