1. What is an absolutely convergent series? What can you say about such a series? How can it compare to a conditionally convergent series? Give an example of an absolutely converging series and a conditionally converging series. An absolutely convergent series is a series with terms that may be positive or negative, but the sum of the absolute value of the terms converge to a real number. According to the notes from our supplemental resources, "Every absolutely convergent series is also convergent. If a series is convergent, but not absolutely convergent, then it is called conditionally convergent." In other words, any series that converges, but is not absolutely convergent, is considered conditionally convergent. (-1)n-1 An example of an absolutely converging series from the book: _1 =1- + 42 + . . n2 32 This the absolute value of the terms are convergent by the p-series test. An example of a conditionally convergent series: _=1 (-1) *2 -1 n =1- 1 + ... The alternating harmonic series is conditionally convergent. If you were to take the absolute value of those terms, it would give you the divergent harmonic series...which diverges. 4. Investigate the Riemann Zeta Function as a series. What is the Riemann hypothesis? According to mathworld.wolfram Ex, The Reimann Zeta Function has uses in physics in regards to definite integration and has close ties with the prime number theorem. Many parts of the theory have been investigated and proven, however there are still important fundamental conjectures, such as the Reimann hypothesis, that are still unproven. The integral of the Reimann zeta function looks like this: 5 (2) = () Jo e-idu The Reimann Hypothesis enters the picture when evaluating the zeros of the Riemann Zeta Function. According to mathworld.wolfram Ex, the generalized Reimann hypothesis says that "neither the Riemann zeta function nor any Dirichlet L-series has a zero with real part larger than ." As noted above, this has never been proven