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Cannot compute limit of sum(log(k),k,1,n)/(n*log(n))

asked 2018-11-22 10:31:42 -0500

wissme gravatar image

updated 2018-11-22 11:23:05 -0500

slelievre gravatar image

Sage can compute this limit

limit(sum(log(k), k, 1, n) / (n * log(n)), n=infinity)       # 1

but not this one?

limit(sum(k*log(k), k, 1, n), n=infinity)

Same with k*(log(k))^2. Any hint?

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answered 2018-11-26 08:27:08 -0500

dan_fulea gravatar image

Well, sage can compute / give a simpler formula for

sage: sum(log(k), k, 1, n)
log(factorial(n))

but cannot do something "more exactly" with

sage: sum(k*log(k), k, 1, n)
sum(k*log(k), k, 1, n)

sage: sum(k*(log(k))^2, k, 1, n)
sum(k*log(k)^2, k, 1, n)

... so we have to solve or work as humans. (For instance by taking the last term in each sum...)

sage: limit(sum(k*log(k), k, n, n), n=infinity)
+Infinity

... or by writing down the simultaneously converging integral, which diverges, so produces a ValueError.

sage: integral( k*log(k), k, 1, oo )
ValueError: Integral is divergent.
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Asked: 2018-11-22 10:31:42 -0500

Seen: 105 times

Last updated: Nov 26 '18