Does the series (1/4)^n that goes to infinity with n = 0 converge or diverge?
This is a geometric series with ratio 1/4 so this series converges and the sum is 1/(1-(1/4)) = 4/3.
Fill in the blanks based on the following options: 1 - Overshadowed, invalidated, illuminated 2 - Enhance, obscure, underscore 3 - Plausibility of our hypothesis, certainty of our entitlement, superficiality of our theories 1. It is refreshing to read a book about our planet by an author who does not allow facts to be (1)__________ by politics: well aware of the political disputes about the effects of human activities on climate and biodiversity, this author does not permit them to (2)__________ his comprehensive description of what we know about our biosphere. He emphasizes the enormous gaps in our knowledge, the sparseness of our observations, and the (3)__________, calling attention to the many aspects of planetary evolution that must be better understood before we can accurately diagnose the condition of our planet.
Overshadowed, obscure, and superficiality of our theories. "Invalidated" doesn't quite fit because it implies that politics is completely disagrees with the facts about our planet, which we can see isn't true later on in the sentence. "Illuminated" indicates that politics completely agrees with the facts, and so 1 should be the answer, because it is a balance between "invalidated" which is too strong of a word, and "illuminated," which is just the opposite of what we're looking for. For part 2, "obscure" is the only word that conveys a rational statement. It doesn't make sense for politics to "enhance" the author's description, because this is too positive of a word and we just established a negative relationship between politics and the book about our planet. "Underscore" implies that politics supports the author's ideas and this also doesn't make sense because in the previous sentence we established that politics overshadows these ideas. Therefore, "obscure" is the answer." For part 3, "Superficiality of our ideas" is the best answer because the preceding phrases are all negative. The commonality between the first two phrases is that they seem to point our the limitations and faults of humankind, which "superficiality of our ideas" matches, while more self-congratulatory words like "certainty" and "plausibility" do not.
Use a special Gaussian surface around an infinite line charge to find the electric field of the line charge as a function of distance.
A special Gaussian surface has to be shaped such that the electric field will be either parallel or perpendicular to the surface normal and it also has to be constant over that surface. Since we can logically determine that the electric field should be radial for a line charge, we can use a cylinder with its axis along the line for our special Gaussian surface. We can now use Gauss' Law to find the electric field. In the equation, you can pull E outside the integral since the electric field has to be constant over the curved surface, and then denote radius as r and length as L. We'll have E(2pi*r*L) = q/epsilon naught, and since the charge enclosed = the linear charge density * the enclosed length of the line, we can substitute that in, and with some algebra we can rearrange the equation to solve for our electric field and that will be our answer.