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I am studying the book Gamma by Julian Havil and there are two equations involving Bernoulli Numbers stated without proofs: $$\sum_{k=1}^{n}k^m = \frac1{m+1} \sum_{k=1}^{m} {{m+1}\choose k-1} B_{k-1}n^{m-k+1} $$ and $$\sum f(k) = \int f(x)dx + \frac12 (f(1) + f(n)) + \sum \dfrac{B_{2k}}{(2k)!} (f^{2k-1}(n)+f^{2k-1}(1))+R.$$

Is there any good self-learnable explanatory books includes proofs of those two mentioned equations and perhaps more alike equations?

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    Graham, Knuth, Potashnik "Concrete Mathematics". – Andrew Aug 01 '18 at 06:31
  • @Andrew, it doesn't include the second equation at all and also not very readable (like e.g Apostol's ANT).. –  Aug 01 '18 at 14:32
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    The second is in the book as "9.5 Euler’s summation formula" – Andrew Aug 01 '18 at 19:03
  • I find the book Concrete Mathematics very readable: I read it before entering college (age 17 or so) and though I skipped sections I couldn't understand, it was very enjoyable and I learned a lot. You have to read it carefully and get into the spirit to enjoy it but it's really much better than treatments elsewhere. – ShreevatsaR Aug 20 '18 at 22:12

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