Processing math: 46%

Saturday, 28 July 2018




BN(x)=Nn=1cosnx
sin(A+B)sin(AB)=sinAcosBcosAsinBsinAcosB+cosAsinB=2sinBcosA

sin(x/2)Nn=1cos(nx)=12Nn=1(sin(nx+x/2)sin(nxx/2))

כמעט הכל מצטמצם ונשאר:

2sin(x/2)Nn=1cos(nx)=sin(Nx+x/2)sin(xx/2)

Nn=1cos(nx)=sin(Nx+x/2)sin(x/2)2sin(x/2)

Saturday, 21 July 2018



Convergence to zero of a function whose improper integral converges.


It is well known that if a series n=1an converges, then an0 as n. The reason is that since the sequence of partial sums SN=Nn=1an  is a convergent sequence of real numbers, it must be Cauchy, so in particular |Sn|=|Sn+1Sn| can be made smaller than any given ε>0, provided n is sufficiently large.
A similar argument does not hold for convergent integrals. Contrary to this situation, if f is a function for which the improper Riemann integral af(x)dx converges, one cannot deduce that lim. Although one still has a Cauchy condition, namely, that the integrals \int_{\alpha}^{\beta}f(x)\,dx are small if \alpha,\beta are sufficiently large, there is enough freedom to form spikes on the graph of f, which do not contribute much to the integral. Such "spikes" will necessarily cause the function not to be uniformly continuous. If, however, the function is uniformly continuous, then \lim_{x\to\infty}f(x)=0, for if this were not the case, then for some \varepsilon>0 we could find an infinite sequence \{x_n\}_{n=1}^{\infty} such that |x_n-x_{n+1}|>1, say, for which |f(x_n)|>\varepsilon for all n. Since f is uniformly continuous, there exists 0<\delta < 1, such that for all n, if |x-x_n|<\delta, then |f(x_n)-f(x)|<\varepsilon/2. In particular,
\left| \int_{x_n-\delta}^{x_n+\delta}f(x)\,dx\right|\geq \delta\varepsilon\quad\hbox{for all $n$},
contradicting the Cauchy criterion.