Company
Mathematica Solutions to the ISSAC '97 Systems Challenge

Wolfram Research, Inc.


Problem 7

What is [Graphics:ISSACChallengegr122.gif] to 9 significant digits?

[Graphics:ISSACChallengegr123.gif]
[Graphics:ISSACChallengegr124.gif]

Result
[Graphics:ISSACChallengegr125.gif]


Method 1: Calculate the first nonvanishing term in the series expansion.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr126.gif]

We calculate the first nonvanishing terms in the series for f(x) and g(x) at [Graphics:ISSACChallengegr129.gif].

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr130.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr131.gif]

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr132.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr133.gif]

The limit follows easily.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr134.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr135.gif]

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr136.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr137.gif]


Method 2: Calculate the limit numerically.

We approach the origin with the sequence [Graphics:ISSACChallengegr138.gif], stopping when we reach a stationary result.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr139.gif]

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr140.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr141.gif]