From: Severian Subject: Re: Help with Series Sum Date: Wed, 04 Apr 2001 15:11:24 +0100 Newsgroups: sci.math Summary: Abel's theorem: convergent series as limits of power series Wade Ramey wrote: > > In article <3ACAC9CC.86@pointecom.net>, > Lynn Killingbeck wrote: > > > Try looking up (or deriving) the Taylor series for ln(1+x) or ln(x) in > > most any reference, paying attention to the region of convergence. This > > is one end of the region. > > Just a quick question: We know ln(1+x) = x - x^2/2 + x^3/3 - for x in > (-1,1), and we know 1 - 1/2 + 1/3 - ... converges, but how do we know the > latter converges to ln(2)? Is this a rhetorical question, or are you unfamiliar with the theorem of Abel which states that if the series sum a_n is convergent, then its sum equals lim_{x -> 1-} sum s_n x^n? -- Severian --------------------------------------------------------------------- "There is no limit to stupidity. Space itself is said to be bounded by its own curvature, but stupidity continues beyond infinity." Gene Wolfe, _The Citadel of the Autarch_ ============================================================================== From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: Help with Series Sum Date: 8 Apr 2001 20:23:08 -0400 Newsgroups: sci.math In article , Wade Ramey wrote: :In article <3ACB2B8C.BA6A0BC0@matachin.fsnet.co.uk>, : Severian wrote: : :> Wade Ramey wrote: : :> > Just a quick question: We know ln(1+x) = x - x^2/2 + x^3/3 - :> > for x in (-1,1), and we know 1 - 1/2 + 1/3 - ... converges, :> > but how do we know the latter converges to ln(2)? :> :> Is this a rhetorical question, or are you unfamiliar with the :> theorem of Abel which states that if the series sum a_n is :> convergent, then its sum equals lim_{x -> 1-} sum s_n x^n? : :No, I know Abel's result. I just thought I'd toss out the :question for those readers who might not have thought about it :before. : :Wade To some advanced readers' surprise, the ln(2) series can be proved directly, long before Abel's Theorem is introduced. Look at the partial sums: 1 - 1/2 + 1/3 - ... + (-1)^(n-1) / n = int[0 to 1] (1 - x + x^2 - ... + (-1^(n-1)*x^(n-1)) dx = int[0 to 1] (1 - (-1)^n * x^n) / (1 + x) dx = int[0 to 1] dx/(1 + x) - int[0 to 1] (-1)^n * x^n / (1 + x) dx = ln(2) + R(n) Observe that 0 < 1/(1 + x) <= 1, so that abs(R(n)) <= 1/(n+1) and the convergence of 1 - 1/2 + 1/3 - ... to ln(2) is proved. Exercise: Show that (Leibniz's series) 1 - 1/3 + 1/5 - 1/7 + ... = pi/4 , again without Abel's Theorem. Cheers, ZVK(Slavek).