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 | | From: | Zaeem Burq | | Subject: | Re: Convergent? | | Date: | Sun, 23 Jan 2005 23:53:13 +0000 (UTC) |
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 | On 23 Jan 2005, Eamon Warnock wrote:
>Prove the convergence or otherwise of
>
>1/1 + 1/2(1+log2) + 1/3(1+log3)(1+log(1+log3)) +
>1/4(1+log4)(1+log(1+log4))(1+log(1+log(1+log4))) + ...
Dear Eamon,
You may wish to experiment with Mathematica: Define the function g[n] as:
h[x_] := 1 + Log[x];
g[n_Integer] := Product[Nest[h, n , m - 1 ], {m, 2, n}]/n
Then g[n] defines the n'th term in your series.
After spending a short time on this, it seems to me that the series converges to + \infty. This is because for each summand, the factor (1 + Log[n])/n converges to 0 at a slower rate than 1/n, whereas all the other factors converge to 1 from above.
Best, Zaeem.
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