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Is there any general formula to sum following series:

$$S = 1^1 + 2^2 + 3^3 + \dotsb+(n - 1)^{n - 1} + n^n, n \in N$$

I mean for $S = f(n)$, is there a formula to compute $f(n)$?

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It seems to me a possible duplicate, but I don't remember exactly. – vesszabo Jan 14 '13 at 19:01
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Nope, sorry. This is equivalent to the integral of $x^x$. Faulhaber's formula is about polynomials. – GregRos Jan 15 '13 at 5:43
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In general, if there isn't an elementary integral for a function there will not be a partial sum formula either. – GregRos Jan 15 '13 at 5:45
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I found the first few values (for $n = 1, \dots , 4$), and this sequence (not surprisingly) appears in OEIS. You can find more information there. – JavaMan Jan 15 '13 at 5:52
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From simple graphing, I've managed to find really good bounds for your function:

$$\log_{10}(f(n))\le\frac{\ln(n^n)}{2.301}$$

I have no guarantee that my inequalities will hold true, but they most certainly hold true from graphing for $20\le n\le143$, which is decently large.

If your wondering how I derived this, I took a guess at how fast $f(n)$ grew and attempted to derive a formula with equivalent growth, seeing that $n^n<f(n)<(n+1)^{n+1}$, we can see how fast it grows, and I just took the log of it to keep numbers manageable.

From a nice article, I took the most understandable parts of it to see that for $n>2$,

$$n^n\left(\frac{4n-3}{4n-4}\right)\le f(n)\le n^n\left(\frac{2+e(n-1)}{e(n-1)}\right)$$

At proposition 2.1.

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