tailor series

[bhuvaneshnick]

question: tailor series of tan((pi/4)+x )
given answer is: 1+2x+4((x^3)/2!)..............

actually is tailor series f(x)+f'(x)((x-a)/1!+.......

so by the above the second term should be 2(x-(pi/4))/1! rather that 2x.

am i right

 

[Ishuda]

question: tailor series of tan((pi/2)+x )
given answer is: 1+2x+4((x^3)/2!)..............

actually is tailor series f(x)+f'(x)((x-a)/1!+.......

so by the above the second term should be 2(x-(pi/4))/1! rather that 2x.

am i right?

Certainly something is strange if you take things at face value, i.e. the general form of the Taylor [not tailor] series would indicate that the expansion was about zero, but the tan(pi/2) is infinity not 1 as indicated in the given series. Even if we reduce this to the cotan function and expand about x=0 we get
tan(pi/2 + x) = sin(pi/2 + x)/cos(pi/2 + x) = -cos(x)/sin(x) = - cot(x) ~ -x^-1 + (1/3) x + (1/45) x³ + ...
which does not correspond to the given series.

As you say, the constant value of 1 in the given series indicates an expansion about a = (n pi - pi/4). Using n=0, it would be as you say (after correcting the typo), the second term should be 2(x+(pi/4))/1! rather that 2x.

 

[bhuvaneshnick]

sorry its tan((pi/4)+x) not tan((pi/2)+x)

 

[bhuvaneshnick]

i would like to know what is the centre value(i.e 'a') for tan((pi/4)+x),since it is not given in the question.Thank you

 

[Ishuda]

i would like to know what is the centre value(i.e 'a') for tan((pi/4)+x),since it is not given in the question.Thank you

If it is the expansion of tan(x + \(\displaystyle \frac{\pi}{4}\)) for the given series, then the expansion point is 0, i.e. x = x - a so a = 0 or, as we did for the other series, tan(a + \(\displaystyle \frac{\pi}{4}\)) = 1 so a = \(\displaystyle n \pi\) and we can choose n = 0.

BTW: note that if we let t = x - \(\displaystyle \frac{\pi}{4}\), then
tan( x + \(\displaystyle \frac{\pi}{4}\)) = 1 + 2 x + 4 x³ /2! + ...
becomes
tan( t + \(\displaystyle \frac{\pi}{2}\)) = 1 + 2 (t + \(\displaystyle \frac{\pi}{4}\)) + 4 (t + \(\displaystyle \frac{\pi}{4})^3\) / 2! + ...