Advanced Calculus fi..

Chapter 6 Infinite Seriesi ,Figure 6.4Proof of integral test for convergence.THEOREM 15 The harmonic series of order p:I ' >tconverges for p > 1 and diverges for p 5 1.hooj The nth term fails to converge to o when p 'zff $%'gat thb'snies surelydiverges when p ( 0. For p > 0 the integral text can be applied, with f (x) = 1 /xP.Now for p # 1,This limit exists and has the value 1 /(p - 1) for p > 1. For p < 1 the integral diverges,and for p = 1,Ilm : b-tm-dx = lim logb = oo,so that again there is divergence. The theorem is now established.When p = 1, one obtains the divergent seriescommonly called the harmonic series; the connection with harmony (that is, music)will be pointed out in connection with the Fourier series in the next chapter. Forgeneral (even complex) values of p the series of Theorem 15 defines a function{(p), the zeta-function of Riemann.THEOREM 16The geometric seriesconverges for - 1 -c r < 1 :and diverges for Ir 1 > 1.