lpetrich
Contributor
Having handled the Taylor part, I turn to the Swift part.
Let's look at infinite-series convergence. One of the series posted by SLD was the binomial series:
Taking the Taylor series of the LHS around x = 0 gives the RHS.
For nonnegative-integer p, one gets the binomial theorem.
For p = -1, one gets
\( \displaystyle{ \frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - x^5 + \cdots } \)
For x = -1 one gets 1/0 = 1 + 1 + 1 + 1 + 1 + ... which obviously diverges.
But for x = 1, one gets 1/2 = 1 - 1 + 1 - 1 + ... which is indeterminate.
For x = 2 one gets 1/3 = 1 - 2 + 4 - 8 + 16 - 32 + ... which diverges
For x = -2 one gets -1 = 1 + 2 + 4 + 8 + 16 + 32 + ... which diverges
So the binomial series only gives meaningful results inside its radius of convergence: |x| < 1.
Let's look at infinite-series convergence. One of the series posted by SLD was the binomial series:
- Binomial theorem and Binomial Theorem -- from Wolfram MathWorld
- Binomial series and Binomial Series -- from Wolfram MathWorld
Taking the Taylor series of the LHS around x = 0 gives the RHS.
For nonnegative-integer p, one gets the binomial theorem.
For p = -1, one gets
\( \displaystyle{ \frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - x^5 + \cdots } \)
For x = -1 one gets 1/0 = 1 + 1 + 1 + 1 + 1 + ... which obviously diverges.
But for x = 1, one gets 1/2 = 1 - 1 + 1 - 1 + ... which is indeterminate.
For x = 2 one gets 1/3 = 1 - 2 + 4 - 8 + 16 - 32 + ... which diverges
For x = -2 one gets -1 = 1 + 2 + 4 + 8 + 16 + 32 + ... which diverges
So the binomial series only gives meaningful results inside its radius of convergence: |x| < 1.