- Determine whether the following series converge absolutely, conditionally, or not
at all and give reasons for your answers.
- ∑
_{n=1}^{∞}^{n} - ∑
_{n=1}^{∞}^{n} - ∑
_{n=1}^{∞}^{n} - ∑
_{n=1}^{∞}^{n} - ∑
_{n=1}^{∞}^{n} - ∑
_{n=1}^{∞}^{n} - ∑
_{n=1}^{∞}^{n} - ∑
_{n=1}^{∞}^{n} - ∑
_{n=1}^{∞}^{n}

- ∑
- Suppose ∑
_{n=1}^{∞}a_{n}converges. Can the same thing be said about ∑_{n=1}^{∞}a_{n}^{2}? Explain. - A person says a series converges conditionally by the ratio test. Explain why his statement is total nonsense.
- A person says a series diverges by the alternating series test. Explain why his statement is total nonsense.
- 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, can you conclude the sum, ∑_{n=1}^{∞}a_{n}b_{n}converges? - If ∑
_{n=1}^{∞}a_{n}converges absolutely, and b_{n}is bounded, can you conclude ∑_{n=1}^{∞}a_{n}b_{n}converges? What if it is only the case that ∑_{n=1}^{∞}a_{n}converges? - Prove Theorem 5.3.4. Hint: For ∑
_{n=1}^{∞}^{n}b_{n}, show the odd partial sums are all no larger than ∑_{n=1}^{∞}^{n}b_{n}and are increasing while the even partial sums are at least as large as ∑_{n=1}^{∞}^{n}b_{n}and are decreasing. Use this to give another proof of the alternating series test. If you have trouble, see most standard calculus books. - Use Theorem 5.3.4 in the following alternating series to tell how large n must be so that
is no larger than the given number.
- ∑
_{k=1}^{∞}^{k},.001 - ∑
_{k=1}^{∞}^{k},.001 - ∑
_{k=1}^{∞}^{k−1},.001

- ∑
- Consider the series ∑
_{k=0}^{∞}^{n}. Show this series converges and so it makes sense to write^{2}. What about the Cauchy product of this series? Does it even converge? What does this mean about using algebra on infinite sums as though they were finite sums? - Verify Theorem 5.4.8 on the two series ∑
_{k=0}^{∞}2^{−k}and ∑_{k=0}^{∞}3^{−k}. - You can define infinite series of complex numbers in exactly the same way as infinite
series of real numbers. That is w = ∑
_{k=1}^{∞}z_{k}means: For every ε > 0 there exists N such that if n ≥ N, then< ε. Here the absolute value is the one which applies to complex numbers. That is,=. Show that ifis a decreasing sequence of nonnegative numbers with the property that lim_{n→∞}a_{n}= 0 and if ω is any complex number which is not equal to 1 but which satisfies= 1 , then ∑_{n=1}^{∞}ω^{n}a_{n}must converge. Note a sequence of complex numbers,converges to a + ib if and only if a_{n}→ a and b_{n}→ b. See Problem 6 on Page 138. There are quite a few things in this problem you should think about. - Suppose lim
_{k→∞}s_{k}= s. Show it follows lim_{n→∞}∑_{k=1}^{n}s_{k}= s. - Using Problem 13 show that if ∑
_{j=1}^{∞}converges, then it follows - Show that if
_{i=1}^{∞}are the prime numbers, then ∑_{i=1}^{∞}= ∞. That is, there are enough primes that the sum of their reciprocals diverges. Hint: Let ϕdenote the number of primes less than equal to n. Then explain whyand consequently why lim

_{n→∞}ϕ= ∞ and ∑_{i=1}^{∞}= ∞. You supply the details for the above.

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