Here is quiz 7.
Problem 1 asks to write out the first few terms of the following series to show how the series starts, and then find the sum of the series:
Let denote the sum of the first terms of the series. Then, for example,
The series is geometric, i.e., it has the form In this case, and Since the series converges, and adds up to
Problem 2 asks to explain why the following series converges or diverges, and if it converges, find its sum:
The series diverges. Perhaps the easiest way of checking this is by using the -th term test: If a series converges, then So, if then diverges. In this case,
so the series diverges.
Problem 3 asks to express the following number as the ratio of two integers:
To do this, simply note that
and that (except for the first term) this is a geometric series,
with and Hence, the series adds up to
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