Recent content by 22/7

When is the aforementioned reversal valid? (For example, is continuity in the neighborhood of the limit sufficient?) Also, how would one prove this? In the example that I am analysing (that of the calculation of a second partial derivative), the function has a removable discontinuity at the...

Hints (a) Use the same Taylor's Theorem from real analysis (works the same way for analytic functions) (b) Subtract F and G from each other and compose the series into one series. Use the result from (a) so that a-b=0 for all n.

I have noticed that, in the theory of divergent series and divergent integrals, \sin ( \infty ) = 0. For example [SIZE="7"][SIZE="6"]\sum_{n=0}^\infty \ (-1)^n = 1 - 1 + 1 - 1 ... = \frac{1}{1-(-1)} = \frac{1}{2} but [SIZE="7"]\sum_{n=0}^k \ (-1)^n = \frac{ - (-1)^{n} -...

Using Bernoulli's approach to this integral (x^x=e^(x*ln(x))=1+x*ln(x)+x^2*(ln(x))^2/2...), I found an infinite sum that converges very quickly but requires the computation of the gamma function and the upper incomplete gamma function. A special case in which the two cancel is the integral...

I think that you may be thinking of when a constant is added to a function. That is when the constant may be ignored in differentiation.

there is one place where the integral can be evaluated explicitly that I know of. \int-log[-log[x]]dx=Euler Constant (.577...) This follows from differentionation the gamma function in its product and integral forms and making a change of variables.