Alternating series test: basic illustration
(a) converges by the AST.
Notice that diverges as a -series with .
Therefore the first series converges conditionally.
(b) converges by the AST.
Notice the funny notation: .
This series converges absolutely because , which is a -series with .
Approximating pi
The Taylor series for is given by:
Use this series to approximate with an error less than .
Solution
(1) The main idea is to use and thus . Therefore:
and thus:
(2) Write for the error of the approximation, meaning .
By the AST error formula, we have .
We desire such that . Therefore, calculate such that , and then we will know:
(3) The general term is . Plug in in place of to find . Now solve:
We conclude that at least terms are necessary to be confident (by the error formula) that the approximation of is accurate to within .