- #1
ddddd28
- 73
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I tried to find the integral of x^m using the definition of Riemann summation. Everything went smoothly until the limit of ∑n=1kn^m divided by k^( m+1), when k approached infinity, showed up.
It is clear that it approaches to 1/m+1, but it has to be proved, of course.
One could induce that fact easily by substituting the closed formula of the sum of powers, involving the Bernoulli numbers, but I think it might be out of the blue, considering that this formula is much more advanced than the initial integral.
At any rate, I had several ideas, which all of a them basically fell short, like trying to rearrange the terms of the sum, so maybe induction can be used or proving that the degree of the sum is m+1 and that the leading coefficient is 1/m+1, so when taking the limit, all the terms except for the first one will go to zero, without knowing them explicitly.
I managed to prove that the degree must be greater than m and less or equal than m+1, but then got stuck.
Any suggestions on how to approach the limit?
It is clear that it approaches to 1/m+1, but it has to be proved, of course.
One could induce that fact easily by substituting the closed formula of the sum of powers, involving the Bernoulli numbers, but I think it might be out of the blue, considering that this formula is much more advanced than the initial integral.
At any rate, I had several ideas, which all of a them basically fell short, like trying to rearrange the terms of the sum, so maybe induction can be used or proving that the degree of the sum is m+1 and that the leading coefficient is 1/m+1, so when taking the limit, all the terms except for the first one will go to zero, without knowing them explicitly.
I managed to prove that the degree must be greater than m and less or equal than m+1, but then got stuck.
Any suggestions on how to approach the limit?