- Determine whether the following series converge and give reasons for your answers.
- ∑
_{n=1}^{∞} - ∑
_{n=1}^{∞} - ∑
_{n=1}^{∞} - ∑
_{n=1}^{∞} - ∑
_{n=1}^{∞} - ∑
_{n=1}^{∞}^{n} - ∑
_{n=1}^{∞}^{n2 }

- ∑
- Determine whether the following series converge and give reasons for your
answers.
- ∑
_{n=1}^{∞} - ∑
_{n=1}^{∞} - ∑
_{n=1}^{∞} - ∑
_{n=1}^{∞}

- ∑
- Find the exact values of the following infinite series if they converge.
- ∑
_{k=3}^{∞} - ∑
_{k=1}^{∞} - ∑
_{k=3}^{∞} - ∑
_{k=1}^{N}

- ∑
- Suppose ∑
_{k=1}^{∞}a_{k}converges and each a_{k}≥ 0. Does it follow that ∑_{k=1}^{∞}a_{k}^{2}also converges? - Find a series which diverges using one test but converges using another if possible. If this is not possible, tell why.
- If ∑
_{n=1}^{∞}a_{n}and ∑_{n=1}^{∞}b_{n}both converge and a_{n},b_{n}are nonnegative, can you conclude the sum, ∑_{n=1}^{∞}a_{n}b_{n}converges? - If ∑
_{n=1}^{∞}a_{n}converges and a_{n}≥ 0 for all n and b_{n}is bounded, can you conclude ∑_{n=1}^{∞}a_{n}b_{n}converges? - Determine the convergence of the series ∑
_{n=1}^{∞}^{−n∕2}. - Is it possible there could exist a decreasing sequence of positive numbers,
such that lim
_{n→∞}a_{n}= 0 but ∑_{n=1}^{∞}converges? (This seems to be a fairly difficult problem.) Hint: You might do something like this. ShowNext use a limit comparison test with

Go ahead and use what you learned in calculus about ln and any other techniques for finding limits. These things will be discussed better later on, but you have seen them in calculus so this is a little review.

- Suppose ∑
a
_{n}converges conditionally and each a_{n}is real. Show it is possible to add the series in some order such that the result converges to 13. Then show it is possible to add the series in another order so that the result converges to 7. Thus there is no generalization of the commutative law for conditionally convergent infinite series. Hint: To see how to proceed, consider Example 5.1.11.

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