Asymptotic results of the remainder in a Ramanujan series for 1/π

Ramanujan gave 17 series for 1/pi. We mention here 1 root = 2 pi 2 1103 992+ 27493 996 1 2 1 middot 3 42 + 53883 9910 1 middot 3 1 middot 3 middot 5 middot 7 2 middot 4 22 middot 82 + ... = n-ary sumation infinity k=0 1 (4k)! 26390k + 1103 28k k!4 994k+2. Ramanujan has pointed out that this series is extremely rapidly convergent. Now let Rn = 1 root 2 pi 2 n-ary sumation n k=0 1 (4k)! 26390k + 1103 28k k!4 994k+2. In this paper, we obtain an asymptotic expansion of the remainder Rn. More precisely, we prove Rn similar to 1 (4n)! (n!)4 n 28n 994n+2 { 1 } 3640 - 3035509 24114272000n + 27421461880263 159752229145600000n2 - ... , n -> infinity. Moreover, we give a recursive relation for determining the coefficients of n1k (k >= 0) in expansion. Then we obtain the upper and lower bounds of Rn. As an application, we give the approximate value of pi. Also, we consider a series for 1/pi due to Chudnovsky brothers.(c) 2022 Elsevier GmbH. All rights reserved.

Automated summary

In this paper, we study series for 1/π and obtain the asymptotic expansion of the remainder, as well as the recursive relation for the coefficients in the expansion. We also provide the upper and lower bounds of Rn and the approximate value of π.