Week 2 - Series
- In general, if we try to add the terms of an infinite sequence we get an expression of the form
- which is called an infinite series (or just a series) and is denoted, for short, by the symbol
1.1. Convergent and Divergent
- Given a series , let denote its nth partial sum:
- If the sequence is convergent and exists as a real number, then the series is called convergent and we write
- The number s is called the sum of the series. If the sequence is divergent, then the series is called divergent.
2. Geometric Series
- Each term is obtained from the preceding one by multiplying it by the common ratio r.
- If , then . Since doesn't exist, the geometric series diverges in this case.
- If , we have
- Subtracting these equations, we get
- If , then , so
- Summarize the results The geometric series is convergent if and its sum is If , the geometric series is divergent.
- Another way to get the conclusion ():
- This figure provides a geometric demonstration of the result in the example. If the triangles are constructed as shown and s is the sum of the series, then, by similar triangles,
- Notice the formula is , so here
3. Telescoping Series
- A telescoping series is a series whose partial sums eventually only have a fixed number of terms after cancellation.
- For example, the series
- (the series of reciprocals of pronic numbers) simplifies as
- From Wikipedia
4. Harmonic Series
- is divergent.
- Solution: For this particular series it’s convenient to consider the partial sums and show that they become large.
- Similarly, , and in general
- If the series is convergent, then .
- If does not exist or if , then the series is divergent.
- If and are convergent series, then so are the series (where c is a constant), , and , and
6. The Comparison Tests
- The Comparison Test Suppose that and are series with positive terms.
- If is convergent and for all n, then is also convergent.
- If is divergent and for all n, then is also divergent.
- Example: Does converges?
- Let's start with n = 2, which should get the same conclusion.
7. Cauchy Condensation
- The sequence decreasing and . The series converges if and only if converges.
- , so is non-decreasing.
- Same as the Harmonic Series part, we can get:
- Same example: Does converges?
- First way -> Use Telescoping Series:
- converge, iff converge.
- => if , then converge.
- which is a telescoping series.
- So converge => converge => converge
- Second way -> Use Conchy Condensation:
- The sequence decreasing and , So:
- With Geometric Series