- Determine whether the following series converge absolutely, conditionally, or
not at all and give reasons for your answers.
(a) ∑

_{n=1}^{∞}^{n}(b) ∑_{n=1}^{∞}^{n}(c) ∑_{n=1}^{∞}^{n}(d) ∑_{n=1}^{∞}^{n}(e) ∑_{n=1}^{∞}^{n}(f) ∑_{n=1}^{∞}^{n}(g) ∑_{n=1}^{∞}^{n}(h) ∑_{n=1}^{∞}^{n}(i) ∑_{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, does ∑_{n=1}^{∞}a_{n}b_{n}converge? - If ∑
_{n=1}^{∞}a_{n}converges absolutely, and b_{n}is bounded, does ∑_{n=1}^{∞}a_{n}b_{n}converge? What if it is only the case that ∑_{n=1}^{∞}a_{n}converges? - Prove Theorem 10.4.4. 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 10.4.4 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 10.5.7 on the two series ∑
_{k=0}^{∞}2^{−k}and ∑_{k=0}^{∞}3^{−k}. - All of the above involves only real sums of real numbers. However, 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. 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. Note how this shows that there are infinitely many prime numbers.

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