=== Formulas converging to ''f''(''x''+1) and ''f''(''x''-1)=== :If ''x'', ''f''(''x''), and ''f''(''x''+1) are all real numbers, and ''f'' can be differentiated infinitely many times, then the following series converges to ''f''(''x''+1) :''f''(''x'') + ''f'' '(''x'') + ''f''(double prime)(''x'')/2 + ''f''(triple prime)(''x'')/6 + ''f''(quadruple prime)(''x'')/24... where the numerator of each term is the function differentiated n times and the denominator is n! :A similar series, the only difference is that the terms alternate in signs, converges to ''f''(''x''-1)'' Yes, that's Taylor's_theorem, in special cases. However you'd need more than a Smooth_function for ''f''. :Do you have a similar formula converging to ''f''(''x''+''c'') where ''x'', ''c'', ''f''(''x''), and ''f''(''x''+''c'') are real numbers?? This is what the page on Taylor's theorem goes into, just with some small changes of notation. You can't say that convergence occurs for just ''any'' function; but it does for many of the common functions like Exponential or Sine. Charles Matthews 19:07, 7 Mar 2004 (UTC) Can you name some functions it doesn't work for?? There are Smooth_functions (all derivatives exist) that are not Analytic_functions (power series). See the Smooth_function page for constructions; also discussed at An_infinitely_differentiable_function_that_is_not_analytic. The power series for a function such as log (1+x) has a Radius_of_convergence that is only 1; so it can't be used if |x| > 1. Charles Matthews 08:14, 8 Mar 2004 (UTC) == Formula of a Power Series == I am reading Penrose's "Road To Reality" where he states (but doesn't demonstrate) that 1 + x²+x⁴+x⁶+x⁸+... = (1-x²)^-1. I understand how both of these are different ways of looking at the same function, but how is it possible to get from one to the other ? (Penrose has a site set up for the solutions to the problems in the book, but is 'too busy' to actually post the solutions there...) :That's a Geometric_series on the left, with common ratio x². Charles Matthews 16:23, 20 Nov 2004 (UTC) == Power series with x values to positive powers == Is, for example, $f(x)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; a\_n\; \backslash left(\; x^2\; \backslash right)^n$ a power series? If so, why is this function a power series but not a series with x to a fractional power? Isn't this function like a power series with a $c\_n$ value that depends on x? --Ott0 00:28, 13 December 2005 (UTC) :Ah, got the answer. This series satisfies the form if a_n is 1 for even powers of n and 0 for odd powers on n. --Ott0 00:52, 13 December 2005 (UTC)