Borbin the 🐱

National 😺 Day!
And in two days it's 🎃👻
Be aware, turns a 🐈 into a 🐈‍⬛

My cat reminded me today of the International 😺 Day!
Do not forget the extra treats today!
🐈🐈🐈🐈🐈🐈🐈🐈🐈🐈🐈🐈🐈🐈🐈

My cat asked me for a staircase. After a long design discussion he finally said: "meow"! Project is on!

All parts except the round material were cut from those boards.

The boards were cut in length.

And to thinner size.

The staircase has an angle of 30°

With a fixed vise all the holes in the treads were drilled.

One of the wooden rounds broke and I had to cut one manually. Took longer than expected.

The railing holes are cut with a 30° alignment using one of the cut offs as a guide.

Two parts assembly.

Construction finished. Painting!

Customer is happy!

It is his staircase.

It was in one of my first physics courses that the professor used 22/7 in an equation as an approximation. Within a few steps, he calculated the result on the board, leaving us impressed.

So simple, and yet so close:

$$\frac{22}{7} = 3.\overline{142857}$$

$$\pi = 3.14159265358\ldots$$

$$\frac{22}{7} - \pi \approx 0.00126$$

The math department would not have approved, but we engineers were perfectly fine with it!

For Math126 we were talking about the Taylor series for

It looks difficult at first glance, but it is actually quite simple. Let me explain:

Using this as a base:

The derivative of this term is:

Which gets to:

Now move the nominator in the sum to get the solution:

Time for a small PowerShell script for the Taylor series:

function f([double]$x)
{
    [Math]::Pow($x/(1-$x), 2)
}

function T([double]$x, [int]$k)
{
    0..$k | % { $sum = 0 } { $sum += $_ * [Math]::Pow($x, $_ + 1) } { $sum }
}


[int]$K = 16

for([int]$xi = -5; $xi -le 5; $xi += 1)
{
    [double]$x = $xi / 10

    [PSCustomObject]@{
        'x' = $x
        'f(x)' = f $x
        'T(x)' = T $x $K
    }
}

   x                f(x)                T(x)
   -                ----                ----
-0,5   0,111111111111111   0,111068725585938
-0,4  0,0816326530612245  0,0816318326348185
-0,3  0,0532544378698225  0,0532544328723274
-0,2  0,0277777777777778  0,0277777777741005
-0,1 0,00826446280991736 0,00826446280991734
   0                   0                   0
 0,1  0,0123456790123457  0,0123456790123457
 0,2              0,0625  0,0624999999943475
 0,3   0,183673469387755   0,183673459741776
 0,4   0,444444444444445   0,444442421037629
 0,5                   1   0,999862670898438

The convergence is | x | < 1
T₁₅ is used for this test.

 k             Tk(x)
 -             -----
 0                 0
 1              0,25
 2               0,5
 3            0,6875
 4            0,8125
 5          0,890625
 6            0,9375
 7        0,96484375
 8        0,98046875
 9      0,9892578125
10       0,994140625
11    0,996826171875
12    0,998291015625
13  0,99908447265625
14     0,99951171875
15 0,999740600585938