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Mathematica Solutions to the ISSAC '97 Systems
Challenge
Wolfram Research, Inc.
Problem 7
What is ![[Graphics:ISSACChallengegr122.gif]](ISSACChallengegr122.gif) to 9 significant digits?
![[Graphics:ISSACChallengegr123.gif]](ISSACChallengegr123.gif)
![[Graphics:ISSACChallengegr124.gif]](ISSACChallengegr124.gif)
Result
![[Graphics:ISSACChallengegr125.gif]](ISSACChallengegr125.gif)
Method 1: Calculate the first nonvanishing term in the series
expansion.
![[Graphics:ISSACChallengegr7.gif]](ISSACChallengegr7.gif) ![[Graphics:ISSACChallengegr126.gif]](ISSACChallengegr126.gif)
We calculate the first nonvanishing terms in the series for
f(x) and g(x) at
![[Graphics:ISSACChallengegr129.gif]](ISSACChallengegr129.gif) .
![[Graphics:ISSACChallengegr130.gif]](ISSACChallengegr130.gif)
![[Graphics:ISSACChallengegr131.gif]](ISSACChallengegr131.gif)
![[Graphics:ISSACChallengegr132.gif]](ISSACChallengegr132.gif)
![[Graphics:ISSACChallengegr133.gif]](ISSACChallengegr133.gif)
The limit follows easily.
![[Graphics:ISSACChallengegr134.gif]](ISSACChallengegr134.gif)
![[Graphics:ISSACChallengegr135.gif]](ISSACChallengegr135.gif)
![[Graphics:ISSACChallengegr136.gif]](ISSACChallengegr136.gif)
![[Graphics:ISSACChallengegr137.gif]](ISSACChallengegr137.gif)
Method 2: Calculate the limit numerically.
We approach the origin with the sequence ![[Graphics:ISSACChallengegr138.gif]](ISSACChallengegr138.gif) ,
stopping when we reach a stationary result.
![[Graphics:ISSACChallengegr139.gif]](ISSACChallengegr139.gif)
![[Graphics:ISSACChallengegr140.gif]](ISSACChallengegr140.gif)
![[Graphics:ISSACChallengegr141.gif]](ISSACChallengegr141.gif)
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