Proving Convergence of a Sequence Using Cauchy Criterion

andyfeynman
Messages
10
Reaction score
0

Homework Statement


Show that the sequence {xn}:
xn := (21/1 - 1)2 + (21/2 - 1)2 + ... + (21/n - 1)2 is convergent.

Homework Equations

The Attempt at a Solution


If n > m,
|xn - xm| = (21/n - 1)2 + (21/(n-1) - 1)2 + ... + (21/(m+1) - 1)2
< (21/n)2 + (21/(n-1))2 + ... + (21/(m+1))2
< (21/(m+1))2 + (21/(m+2))2 + ...
= 41/m
Let ɛ > 0. We choose N such that 41/N < ɛ for all n > m > N.
Then |xn - xm| < 41/N < ɛ for all n > m > N.
 
Physics news on Phys.org
andyfeynman said:
Let ɛ > 0. We choose N such that 41/N < ɛ for all n > m > N.
Which N would you choose for ɛ=1?
 
mfb said:
Which N would you choose for ɛ=1?
Just forgot it. I made a very stupid mistake.
But I came up with the idea of letting bk = 21/k - 1.
This means bk2 < 4[k(k-1)] for all k > 2.
Therefore,
xn = 1 + b22 + ... + bn2
< 1 + 4/[1(2-1)] + ... 4/[n(n-1)]
= 1 + 4(1 - 1/2) + ... + 4[1/(n-1) - 1/n]
= 5 - 4/n
< 5
Since xn is monotone increasing and bounded, it is convergent.
But is there any way to do it using the Cauchy criterion?
 
andyfeynman said:
This means bk2 < 4[k(k-1)] for all k > 2.
That step is certainly not trivial.

I would use something like 21/k < 1 + c/k for some c.

Cauchy criterion: Probably, but I don't see how it would help.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
Back
Top