# Continuity, series, inverse function

• Mar 19th 2010, 04:36 PM
GOKILL
Continuity, series, inverse function
1. If $f:\mathbb{R}\mapsto\mathbb{R}$ is monotonic increasing and $f(f(x))$ is continuous, prove that $f(x)$ is continuous!

2. Find the value of $\sum^{\infty}_{k=1} \frac{1}{2^k}\tan (\frac{\pi}{2^{k+2}})$

3. If $f:\mathbb{R}^2\mapsto\mathbb{R}^2\text{ where } f((x_1,x_2))=(x_1+2x_2,2x_1+x_2)\forall x_1,x_2\in\mathbb{R},\text{ then find } f^{-1}$
• Mar 21st 2010, 03:09 AM
Opalg
Quote:

Originally Posted by GOKILL
2. Find the value of $\sum^{\infty}_{k=1} \frac{1}{2^k}\tan (\frac{\pi}{2^{k+2}})$

Let $f(x) = -\sum_{k=1}^\infty\ln\bigl(\cos(x/2^k)\bigr)$. Then $f'(x) = \sum_{k=1}^\infty2^{-k}\tan(x/2^k)$, so we are looking for $f'(\pi/4)$.

To compute f(x), write $\sum_{k=1}^n\ln\bigl(\cos(x/2^k)\bigr) = \ln\Bigl(\prod_{k=1}^n\cos(x/2^k)\Bigr)$. Then use the formula $\cos x\cos y = \tfrac12\bigl(\cos(x+y) + \cos(x-y)\bigr)$ to get

$\cos\tfrac x2\cos\tfrac x4 = \tfrac12\bigl(\cos\tfrac14x +\cos\tfrac34x\bigr),$

$\cos\tfrac x2\cos\tfrac x4\cos\tfrac x8 = \tfrac14\bigl(\cos\tfrac18x +\cos\tfrac38x + \cos\tfrac58x +\cos\tfrac78x\bigr),$

...

$\prod_{k=1}^n\cos(x/2^k) = \frac1{2^{n-1}}\sum_{k=1}^{2^{n-1}} \cos\Bigl(\frac{2k-1}{2^n}x\Bigr)$.

That last line can be viewed as a Riemann sum for $\int_0^1\!\!\!\cos(xt)\,dt = \frac{\sin x}x$. Therefore $f(x) = -\ln\Bigl(\frac{\sin x}x\Bigr)$, and so $f'(x) = x^{-1} - \cot x$.

Finally, $f'(\pi/4) = \frac4\pi-1 \approx 0.2732$.