2. Now consider the series r r8(4n)!(1103 + 26390n) 9801 - 3964n (n!)ª This series was discovered by the extraordinary Indian n=0 mathematician Srinivasa Ramanujan (1887-1920). This series converges to –. (You do NOT need to prove that, and it is much more difficult than finding the sum of the series in problem 1.) This series has been used to compute T to over 17 million digits (which was a world record at the time). V8(4n)!(1103 + 26390n) 9801 · 3964n (n!)4 (a) Use any test for convergence/divergence to show that the series ) n=0 converges. The partial sums for this series are S. = 5 v8(4n)!(1103 + 26390n) 9801 · 3964n (n!)4 (b) %3D n=0 Use a calculator to evaluate and , and write down as many digits as your calculator can display. How many digits are the same as the digits of 7? Note: 7 3.1415926535 8979323846 2643383279...
2. Now consider the series r r8(4n)!(1103 + 26390n) 9801 - 3964n (n!)ª This series was discovered by the extraordinary Indian n=0 mathematician Srinivasa Ramanujan (1887-1920). This series converges to –. (You do NOT need to prove that, and it is much more difficult than finding the sum of the series in problem 1.) This series has been used to compute T to over 17 million digits (which was a world record at the time). V8(4n)!(1103 + 26390n) 9801 · 3964n (n!)4 (a) Use any test for convergence/divergence to show that the series ) n=0 converges. The partial sums for this series are S. = 5 v8(4n)!(1103 + 26390n) 9801 · 3964n (n!)4 (b) %3D n=0 Use a calculator to evaluate and , and write down as many digits as your calculator can display. How many digits are the same as the digits of 7? Note: 7 3.1415926535 8979323846 2643383279...
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Written 8 Q2
Transcribed Image Text:V8(4n)!(1103 + 26390n)
9801 · 3964n (n!)1
2. Now consider the series
This series was discovered by the extraordinary Indian
n=0
mathematician Srinivasa Ramanujan (1887-1920). This series converges to
(You do NOT need to prove that,
and it is much more difficult than finding the sum of the series in problem 1.) This series has been used to compute
T to over 17 million digits (which was a world record at the time).
V8(4n)!(1103 + 26390n)
9801 · 3964n (n!)4
(a)
Use any test for convergence/divergence to show that the series )
n=0
converges.
k
V8(4n)!(1103 + 26390n)
(b)
The partial sums for this series are S,
Σ
9801 · 3964n(n!)4
n=0
Use a calculator to evaluate and , and write down as many digits as your calculator can display. How
many digits are the same as the digits of 7?
Note: 7 3.1415926535 8979323846 2643383279...
Bonus: Watch the movie "The man who knew infinity" (2015) about the life of Srinivasa Ramanujan.
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