POWER SERIES EXPANSION OF TAN about PI/2?

Question

http://www.wolframalpha.com/input/?i=expand+tan%C2%B2z+about+pi%2F2

Anyone know how can we mere mortals get the expansion WolframAlpha got?
Is there a general rule/method to produce power series of functions about their singularities?

Awms A · Accepted Answer

One method:
Find the Taylor series of the function
f(x) =
{ (x-pi/2) * tan(x) . . . if x is not pi/2
{ -1 . . . . if x=pi/2
about pi/2.

Second method:
Use complex analysis.  Since the singularity is isolated, the coefficients in the power series are given by limits of contour integrals around the point z=pi/2.

I'm sure there are others, too.

Anonymous · Answer

Pi / 4 = arctan a million Pi = 4 arctan a million Pi = 4(a million - a million/3 + a million/5 - a million/7 + ...) Pi = 4 - 4/3 + 4/5 - 4/7 + ... **************************************... The sequence does converge, by utilising the alternating sequence try, in spite of the fact that it converges very slowly. All that became into asked for became into an expression, not a superb one. :-)

? · Answer

Theres a formula for it.

If i can recall its:
f^n(a) (x-a) / n!

f^n where n symbolizes what derivative you are on.

and where a is what you are expanding about

i think thats what you are looking for

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POWER SERIES EXPANSION OF TAN about PI/2?

3 Answers