pradyumna1974
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- Aug 11, 2020
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What is problem in the following derivation ?
Problem Statement: what is the value of (ꝏ/ꝏ) ?
As per Wikipedia article on Factorial (https://en.wikipedia.org/wiki/Factorial) , if we consider the original formula of Γ function from Euler for computation of Factorial, we can use the following formula.
Π (n) = Γ(n+1) for n – non-negative integer.
So
Γ(1) = Π (0) (i.e. Π represents “Product” of the series, not π = 3.14159)
For any n, Π(n) = 1*2*3*….*(n-1)*n, where n > 0
For n = 0, Π (n) = Π (0) = 0!
Using original formula of Euler for Gamma function,
Γ(z) = Lim (n-> ꝏ) (( n ^ z ) * ( n ! )) / (Π (k=0 to n) (z + k))
when we compute 0!,
0! = Π (0) = Γ(1) = Lim (n=> ꝏ) ((n ^ 1) * (n!)) / Π (k=0 to n)(1+k)
=> Lim (n=> ꝏ) ((n ^ 1) * n!) / ((1+0)* (1+1) * ...* (1+n-1) * (1+n))
=> Lim (n=> ꝏ) ((n ^ 1) * n!) / (n! * (1+n))
=> Lim (n=> ꝏ) n /(1+n)
=> Lim (n=> ꝏ) ((n +1) /n) ^ -1 (Converting to reciprocal version)
=> ((1+ ꝏ)/ꝏ) ^ -1
=> (1/ꝏ + ꝏ/ꝏ) ^ -1 = 1 (where ꝏ Є R and ꝏ is unbounded positive infinity).
=> (1/ꝏ + ꝏ/ꝏ) = 1 ^ 1/-1
=> (1/ꝏ + ꝏ/ꝏ) = 1 ^ - 1
=> (1/ꝏ + ꝏ/ꝏ) = 1/ (1 ^ 1) = 1/1 = 1
1 = 0! => (1/ꝏ + ꝏ/ꝏ) = 1
This is an interesting situation where Left Hand Side (LHS) = (1/ꝏ + ꝏ/ꝏ) is equal to Right Hand Side (RHS) = 1 (a constant integer)
whereas LHS contains two indeterminate forms is 1/ꝏ and ꝏ/ꝏ, when added , the resultant sum is 1.
To simplify this further, applying limits on both sides,
Lim (n=> ꝏ) (1/n + n/n) = Lim (n=>ꝏ) 1
=> Lim (n=>ꝏ) (1/n) + Lim (n=>ꝏ)(n/n) = 1
=> Lim (n=>ꝏ) (n/n) = 1 – Lim (n=>ꝏ) (1/n)
By applying standard calculus formula, we can conclude that Lim (n=>ꝏ) (1/n) = 0
=> Lim (n=>ꝏ) (n/n) = 1 - 0
=> ꝏ / ꝏ = 1
Hence we could conclude that ꝏ/ꝏ = 1
Please help me with feedback on this derivation. Thank you all for your future feedback!
P.S. I am mathematics enthusiast and learner. I am not expert, hence I am using this forum to improve my knowledge.
Problem Statement: what is the value of (ꝏ/ꝏ) ?
As per Wikipedia article on Factorial (https://en.wikipedia.org/wiki/Factorial) , if we consider the original formula of Γ function from Euler for computation of Factorial, we can use the following formula.
Π (n) = Γ(n+1) for n – non-negative integer.
So
Γ(1) = Π (0) (i.e. Π represents “Product” of the series, not π = 3.14159)
For any n, Π(n) = 1*2*3*….*(n-1)*n, where n > 0
For n = 0, Π (n) = Π (0) = 0!
Using original formula of Euler for Gamma function,
Γ(z) = Lim (n-> ꝏ) (( n ^ z ) * ( n ! )) / (Π (k=0 to n) (z + k))
when we compute 0!,
0! = Π (0) = Γ(1) = Lim (n=> ꝏ) ((n ^ 1) * (n!)) / Π (k=0 to n)(1+k)
=> Lim (n=> ꝏ) ((n ^ 1) * n!) / ((1+0)* (1+1) * ...* (1+n-1) * (1+n))
=> Lim (n=> ꝏ) ((n ^ 1) * n!) / (n! * (1+n))
=> Lim (n=> ꝏ) n /(1+n)
=> Lim (n=> ꝏ) ((n +1) /n) ^ -1 (Converting to reciprocal version)
=> ((1+ ꝏ)/ꝏ) ^ -1
=> (1/ꝏ + ꝏ/ꝏ) ^ -1 = 1 (where ꝏ Є R and ꝏ is unbounded positive infinity).
=> (1/ꝏ + ꝏ/ꝏ) = 1 ^ 1/-1
=> (1/ꝏ + ꝏ/ꝏ) = 1 ^ - 1
=> (1/ꝏ + ꝏ/ꝏ) = 1/ (1 ^ 1) = 1/1 = 1
1 = 0! => (1/ꝏ + ꝏ/ꝏ) = 1
This is an interesting situation where Left Hand Side (LHS) = (1/ꝏ + ꝏ/ꝏ) is equal to Right Hand Side (RHS) = 1 (a constant integer)
whereas LHS contains two indeterminate forms is 1/ꝏ and ꝏ/ꝏ, when added , the resultant sum is 1.
To simplify this further, applying limits on both sides,
Lim (n=> ꝏ) (1/n + n/n) = Lim (n=>ꝏ) 1
=> Lim (n=>ꝏ) (1/n) + Lim (n=>ꝏ)(n/n) = 1
=> Lim (n=>ꝏ) (n/n) = 1 – Lim (n=>ꝏ) (1/n)
By applying standard calculus formula, we can conclude that Lim (n=>ꝏ) (1/n) = 0
=> Lim (n=>ꝏ) (n/n) = 1 - 0
=> ꝏ / ꝏ = 1
Hence we could conclude that ꝏ/ꝏ = 1
Please help me with feedback on this derivation. Thank you all for your future feedback!
P.S. I am mathematics enthusiast and learner. I am not expert, hence I am using this forum to improve my knowledge.