Hard to define function and Conv/Dive of Integral/Series

[LCKurtz]

When x = n your second one gives -1, so it's still wrong. And if you simplify your first one by multiplying numerator and denominator by 2ⁿyou can write it as

1 + 2ⁿ(x-n)

You should be able to write the second piece similarly. Keeping it simple is always a good idea.

 

[estro]

When x = n your second one gives -1, so it's still wrong. And if you simplify your first one by multiplying numerator and denominator by 2ⁿyou can write it as

1 + 2ⁿ(x-n)

You should be able to write the second piece similarly. Keeping it simple is always a good idea.

But the second one deals with $$x\in (n,n+(1/2)^n]$$ or I missed something?

 

[LCKurtz]

But the second one deals with $$x\in (n,n+(1/2)^n]$$ or I missed something?

Sure, the second one applies to that interval but your formula for it is wrong. You should get 1 when x = n in both of them.

 

[estro]

$$<br /> f(x) = \left\{<br /> \begin{array}{c l}<br /> 2^n(x-n)-1, & \mbox{if } x \in [n-(1/2)^n, n] \mbox{ while } n=min\{ |\lfloor x \rfloor - x|, |\lfloor x+1 \rfloor - x| \}\\<br /> \\<br /> 2^n(n-x)+1, & \mbox{if } x \in (n, n+(1/2)^n] \mbox{ while } n=min\{ |\lfloor x \rfloor - x|, |\lfloor x+1 \rfloor - x| \}\\<br /> \\<br /> 0, & \mbox{otherwise}<br /> \end{array}<br /> \right.<br />$$

 

[LCKurtz]

Haven't we gone over this before:

1. Your function must have a negative slope on the right side of the interval.

2. When you put x = n in either formula, you have to get 1.

If your formula doesn't satisfy both of those, it is wrong. Think about it. I have to leave now for today.

 

[estro]

I believe it was fatigue blunder (embarrassing).

$$<br /> f(x) = \left\{<br /> \begin{array}{c l}<br /> 2^n(x-n)+1, & \mbox{if } x \in [n-(1/2)^n, n] \mbox{ while } n=min\{ |\lfloor x \rfloor - x|, |\lfloor x+1 \rfloor - x| \}\\<br /> \\<br /> 2^n(n-x)+1, & \mbox{if } x \in (n, n+(1/2)^n] \mbox{ while } n=min\{ |\lfloor x \rfloor - x|, |\lfloor x+1 \rfloor - x| \}\\<br /> \\<br /> 0, & \mbox{otherwise}<br /> \end{array}<br /> \right.$$

Many times I have simple idea for a solution fairly fast, but when I try to apply it and actually write equations my intuition is letting me down and I get lost.
This was a good example for it, without your help most likely I would spent a week(it actually happens to me occasionally) crunching this problem even though having the idea.