approximating sum[j=1,k] 1/j^(phi) by integral 1 + int[0,k]1/(1 + x)^(phi) dx: how?

[clarkson]

There's a term as:

. . . . .\(\displaystyle \displaystyle k^{\phi}\, \sum_{j = 1}^{k}\, \dfrac{1}{j^{\phi}}\)

This sum is approximated by the integral:

. . . . .\(\displaystyle \displaystyle k^{\phi}\, \left(1\, +\, \int_0^k\, \dfrac{1}{(1\, +\, x)^{\phi}}\, dx\right)\)

Can someone please tell me how this approximation is made?

 

[stapel]

There's a term as:

. . . . .\(\displaystyle \displaystyle k^{\phi}\, \sum_{j = 1}^{k}\, \dfrac{1}{j^{\phi}}\)

When you say "there's a term", do you mean "in the course of something else that I'm doing, there is a summation that arises"? If so, what is the "something else"? What information does it provide regarding k, j, and \(\displaystyle \, \phi\)? If not, what do you mean?

This sum is approximated by the integral:

. . . . .\(\displaystyle \displaystyle k^{\phi}\, \left(1\, +\, \int_0^k\, \dfrac{1}{(1\, +\, x)^{\phi}}\, dx\right)\)

Is it safe to assume that we can ignore the \(\displaystyle k^{\phi}\) portion of each of the two expressions when considering the approximation?

When you reply, please include a clear listing of your thoughts and efforts so far. Thank you!