Maclaurin Series tan x[br][img]https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series/maclaurin_series_expansion_general_formula.gif[/img][br][br]Deriving the Maclaurin series for tan x is a very simple process. It is more of an exercise in differentiating using the chain rule to find the derivatives. As you can imagine each order of derivative gets larger which is great fun to work out.[img]https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_tanx/maclaurin_series_tanx_01.gif[/img][br][br]The first one is easy because tan 0 = 0.[img]https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_tanx/maclaurin_series_tanx_02.gif[/img][br][br]The first derivative of tan x is very simple as you can see.[img]https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_tanx/maclaurin_series_tanx_03.gif[/img][br][br]For the second derivative, I am using the chain rule.[img]https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_tanx/maclaurin_series_tanx_04.gif[/img][br][br]Using the chain rule for the third derivative of tan x is not so bad and easily manageable. As you can see, there is a pattern here. The odd-numbered derivatives will have the same value as the constant and the even-numbered derivatives will become zero.[img]https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_tanx/maclaurin_series_tanx_05.gif[/img][br][br]As you can see, for the even-numbered derivative there is no constant. All the terms have a tan function in them and therefore they become zero.[img]https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_tanx/maclaurin_series_tanx_06.gif[/img][br][br]The first three are multiplied by1 so they just copy over as shown above. The fifth derivative has a constant of 16 all the rest of the terms have the tan function so they become zero when x = 0.[img]https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_tanx/maclaurin_series_tanx_07.gif[/img][br][br]This looks like a familiar sequence; I remember reading some ancient sutras from the 2nd century BC, with these numbers in them. The next number should therefore be 272.[img]https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_tanx/maclaurin_series_tanx_08.gif[/img][br][br]As you can see, it is a simple matter of substituting the numbers into the general Maclaurin series as shown above. The terms with the zero will obviously cancel which makes it easier.[img]https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_tanx/maclaurin_series_tanx.gif[/img][br][br]This is the Maclaurin series for tan x, and being a tan function, it will have a narrow range just as your calculator does. See if your calculator can calculate tan 90. One of my old vintage calculators used to crash!