5. Find the Laurent series for (z-1) (2+2)(2-3)5 in 0 < 3<5.

3. The series 1 (4n) !(1103 + 26390n) 9801 (n!)*3964n was discovered by Ramanujan in the early 1900s. William Gosper, Jr., used this series to calculate a to an accuracy of more than 17 million digits in the mid-1980s. At the time, that was a world record. Since that time, this series and others by Ramanujan have led mathematicians to find many other series representations for T and 1/т. a. Prove that this series converges.

a) Determine whether each of the series below converges or diverges and explain your reasoning (k!) ³ (3k)!

3 3 3 Prove that .33333... = 1/3 by writing it as 10 + ... and using the geometric series 1000 100