## Possible Downtime and More Random Math

http://www.neildickson.com and http://www.codecortex.com may or may not be down on Sunday or Monday.  If they are, I blame Scotiabank for:

• needing 6-8 business days to make a simple transaction that can be made in less than half a second because of two little things I like to call: the internet and the 21st century
• sending two versions of a credit card within a week or two where activating the wrong one makes them both not work
• needing over a week to make and send a new credit card instead of just creating one on the spot in about 30 seconds as they easily could (just need a punch for the numbers and a magnetic strip writer)
• charging me an outrageous $54 annual fee for the privilege of carrying a piece of plastic around that didn’t work anyway • charing a record high 30% interest rate on the credit card even though I’ve never missed a payment • paying a record low 0.25% interest rate on my savings account, and even then only on amounts over$5,000; less than 1/3 of what RBC pays, and less than 1/4 of what HSBC pays
• working only 9:30am-4:00pm to avoid dealing with customers who have jobs

Anyway, hopefully the sites don’t go down, and here’s some more random math-ish stuff.  Suppose you have a function in terms of its Taylor series about zero (it’s Maclauren series):

$f(x) = \displaystyle\sum_{i=0}^{\infty} a_i x^i$

How can you find the square of the function?  In hind sight, it should have been obvious, but it’s:

$(f(x))^2 = \displaystyle\sum_{i=0}^{\infty} \left(\displaystyle\sum_{j=0}^{i} a_j a_{i-j}\right) x^i$

In fact, more generally, if we also have $g(x) = \sum_{i=0}^{\infty} b_i x^i$:

$f(x)g(x) = \displaystyle\sum_{i=0}^{\infty} \left(\displaystyle\sum_{j=0}^{i} a_j b_{i-j}\right) x^i$

You can also calculate $\displaystyle\frac{f(x)}{g(x)}$ fairly simply:

$\displaystyle\frac{f(x)}{g(x)} = \frac{\displaystyle\sum_{i=0}^{\infty} a_i x^i}{\displaystyle\sum_{i=0}^{\infty} b_i x^i} = \displaystyle\sum_{i=0}^{\infty} c_i x^i$
$\displaystyle\sum_{i=0}^{\infty} a_i x^i = \left(\displaystyle\sum_{i=0}^{\infty} b_i x^i\right)\left(\displaystyle\sum_{i=0}^{\infty} c_i x^i\right)$

$a_0 = b_0 c_0$
$a_1 = b_1 c_0 + b_0 c_1$
$a_2 = b_2 c_0 + b_1 c_1 + b_0 c_2$
$a_3 = b_3 c_0 + b_2 c_1 + b_1 c_2 + b_0 c_3$

$\cdots$

$\begin{bmatrix} b_0 & 0 & 0 & 0 & \cdots \\ b_1 & b_0 & 0 & 0 & \cdots \\ b_2 & b_1 & b_0 & 0 & \cdots \\ b_3 & b_2 & b_1 & b_0 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix} \begin{bmatrix} c_0 \\ c_1 \\ c_2 \\ c_3 \\ \vdots \end{bmatrix} = \begin{bmatrix} a_0 \\ a_1 \\ a_2 \\ a_3 \\ \vdots \end{bmatrix}$

Simply use forward substitution to solve for the values of $c_i$.

It’s much harder to work out what non-integer powers of the Taylor series are, though.  The pattern for $\sqrt{f(x)}$ seems too complex to figure out from 5 terms.  What’d be really cool is to get the series for $f(g(x))$, since then most anything else would be pretty easy to figure out.