Question

3.) Let ak E R with ak > 0 for all k E N. Suppose Σ㎞iak converges. Show that Σί1bk (By definition, for a sequence (ck), we say liCkoo if, for all M ER with Hint: Show that there exists (Ni))ไ1 with N > Nj for all j E N, such that bk there exists a sequence (bk)k of real numbers such that lim converges = oo and M >0, there exists N E N such that ck > M for all k > N.) ak S and let b -jay for N, Sk<N+ Remark: This result shows that there is no "borderline series, that is, no series a of positive terms that gives the slowest rate at which the terms can converge to 0 and still have that the series Σ ak converges.

Answer 1

The potential is, v(x) = -alpha[delta(x + a) + delta(x - a)] the wave function takes the following form psi = A e^ikx + B e^-ikx(x < -a) = C e^ikx + De^-ikx(-a < x < a) = Fe^ikx (x > a) apply continuity of psi at -a. A e^-ika + B e^ika = C e^-ika + De^ika e^+ika[Ae^-2ika + B] = e^+ika[Ce^-2ika + D] let e^-2ika = beta beta + B = betaC + D . . .(1) apply continuity of psi at a.