#### bhobba

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Ok - anyone that has done basic analysis knows the definition of convergence. The series 1-2+3-4+5..... is for example obviously divergent (alternating series test). But wait a minute lets try something tricky and perform a transform on it, (its Borel summation, but that is not really relevant to this).

∑an = ∑(an/n!)n! = ∑ (an/n!)∫t^n*e^-t where the sum is from n=0 to ∞ and the integral also from 0 to ∞. Now suppose ∑|(an*(xt)^n/n!)e^-t| < ∞ then dominated convergence applies and the integral and sum can be reversed, if the integral exists. It usually does for some x (in the example its |x|<1), but in the actual integral for at least x =1, which means it can be considered an analytic continuation to x=1 so ∫∑((an*t^n)/n!)*e^-t can be taken as ∑an. Or you can consider it limit x → 1- if it exists for |x|<1 because, again from dominated convergence, its continuous in x.

Now lets apply this to 1-2+3-4 ....... where the sum is from n =1 to ∞. We have ∫∑(-n*(-t)^n*e^-t)/n! = ∫∑(-(-t)^n*e^-t)/(n-1)! = ∫∑(-(-t)^(n+1)*e^-t)/(n)! = ∫t*e^-t*e^-t = ∫t*e^-2t = 1/4.

Are divergent series sometimes really convergent but simply written in the wrong form?

Added Later:

Made a goof, but fixed it. Thought I could get away without using analytic continuation - just the limit - but I was wrong. Certainly in the example you can use just the limit - but not more generally.

Just as an aside from Wikipedia:

Borel, then an unknown young man, discovered that his summation method gave the 'right' answer for many classical divergent series. He decided to make a pilgrimage to Stockholm to see Mittag-Leffler, who was the recognized lord of complex analysis. Mittag-Leffler listened politely to what Borel had to say and then, placing his hand upon the complete works by Weierstrass, his teacher, he said in Latin, 'The Master forbids it'.

Mark Kac, quoted by Reed & Simon (1978, p. 38)

Looks like Mittag-Leffler had trouble with it, but interestingly Borel wrote a book on Divergent series later, or maybe it should be superficially divergent series.

Thanks

Bill

∑an = ∑(an/n!)n! = ∑ (an/n!)∫t^n*e^-t where the sum is from n=0 to ∞ and the integral also from 0 to ∞. Now suppose ∑|(an*(xt)^n/n!)e^-t| < ∞ then dominated convergence applies and the integral and sum can be reversed, if the integral exists. It usually does for some x (in the example its |x|<1), but in the actual integral for at least x =1, which means it can be considered an analytic continuation to x=1 so ∫∑((an*t^n)/n!)*e^-t can be taken as ∑an. Or you can consider it limit x → 1- if it exists for |x|<1 because, again from dominated convergence, its continuous in x.

Now lets apply this to 1-2+3-4 ....... where the sum is from n =1 to ∞. We have ∫∑(-n*(-t)^n*e^-t)/n! = ∫∑(-(-t)^n*e^-t)/(n-1)! = ∫∑(-(-t)^(n+1)*e^-t)/(n)! = ∫t*e^-t*e^-t = ∫t*e^-2t = 1/4.

Are divergent series sometimes really convergent but simply written in the wrong form?

Added Later:

Made a goof, but fixed it. Thought I could get away without using analytic continuation - just the limit - but I was wrong. Certainly in the example you can use just the limit - but not more generally.

Just as an aside from Wikipedia:

Borel, then an unknown young man, discovered that his summation method gave the 'right' answer for many classical divergent series. He decided to make a pilgrimage to Stockholm to see Mittag-Leffler, who was the recognized lord of complex analysis. Mittag-Leffler listened politely to what Borel had to say and then, placing his hand upon the complete works by Weierstrass, his teacher, he said in Latin, 'The Master forbids it'.

Mark Kac, quoted by Reed & Simon (1978, p. 38)

Looks like Mittag-Leffler had trouble with it, but interestingly Borel wrote a book on Divergent series later, or maybe it should be superficially divergent series.

Thanks

Bill

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