Pure Mathematics D3: Series

12 flashcards · Mathematics · by lakesare

1. When the: sequence converges VS series converges?

   sequence converges - when it converges to some limit series converges - when its sequence of partial sums (sn) converges

2. If an is the sequence an, what is the sequence of partial sums sn?

3. [Notation] What does this shorthand notation mean?

   Note that there is no term a∞ in the series. The symbol ∞ is here used to mean that the subscript n takes every integer value greater than or equal to 1.

4. [Zeno's Paradox] What's (an) and (sn) (sequence of partial sums) does Achilles trace? In one of his paradoxes, Zeno claimed that it is impossible for an object to travel a given distance, since it must first travel half the distance, then half of the remaining distance, then half of what remains, and so on.

   (an) = 1/2 , 1/4 , 1/8 , ....... (sn) = 1/2 , 1/2 + 1/4 , 1/2 + 1/4 + 1/8 , ...

5. [Convergence tests] Name 3 types of convergence tests. You have met a number of tests that can be used to determine whether a series is convergent and a strategy to help you to work out which test is the most appropriate. If all the terms of a series are non-negative, then you may be able to use the Comparison Test or the Limit Comparison Test to compare with a basic series you know to be convergent or divergent; or you may be able to use the Ratio Test. For a series with infinitely many positi…

   The easiest: (an) must be null for the series to converge. Only positive terms: tests where (an) is always positive. ∞ alternating terms: tests where (an) switches pos/neg. If none of steps 1–3 of the strategy gives a result, then you could try using first principles by working directly with the sequence (sn) of partial sums.

6. When is (an) a sequence with non-negative terms?

   When all terms are non-negative.

7. [Basic series] What's surprising about the ∑1/np basic series?

   Harmonic series ∑1/n1 is divergent This series is often called the harmonic series, since its terms are proportional to the lengths of strings that produce harmonic tones in music.Basel problem series ∑1/n2 converges to pi2/6 The problem of finding the exact sum of this series is known as the Basel problem. It was solved by Leonard Euler, and the problem is named after the city of Basel in Switzerland, the hometown of Euler and of the Bernoulli family.Apery's constant series ∑1/n3 converges to…

8. [Basic series] What's the connection between basic sequences basic series?

   It makes sense that if ∑an is convergent, then an converges to 0.

9. [Telescoping series] What's an interesting way to rewrite this expression?

   This is useful for telescoping series.

10. [Telescoping series] Why is this called telescoping series?

    Because of the cancellation of adjacent terms.

11. Write sequence an VS sequence sn together. sn is a sequence of partial sums of an.

12. We know many ways to determine whether our series converges, however name 2 ways to determine WHAT our series converges to. For all the infinite series we have so far considered, it is possible to derive a simple formula for the nth partial sum. For many series, however, this is difficult or even impossible.

    sum of the geometric series telescoping series