Question

Why is it when we find the Taylor series of sinx/x, we are allowed to simply multiply 1/x into the known sin x Taylor series formula, while we have to multiply manually when finding the formula of the Taylor series of (e^x) cos x?

Answer 1

e^x like cos x and sin x can be represented by a Taylor Expansion, therefore in the case of e^(x) cos x you are actually multiplying the two series while 1/x can not be expanded further so you are only multiplying sin x with 1/x

Answer 2

sin(x) = x - x3/3! + x5/5! - x7/7! + ... + xn.sin(n.pi/2)/n!
cos(x) = 1 - x2/2! + x4/4! - x6/6! + ... + xn.cos(n.pi/2)/n!

ex = 1 + x/1! + x2/2! + x3/3! + ... + xn/n!

Answer 3

hmm, so how can I know whether a series can be expanded or not?

Answer 4

Polynomials in general will be a finite series like how 1/x is just the one term. While trying to express non-polynomial functions like sin,cos,log,e^x in a polynomial notation then it expands into an infinite series.

Answer 5

how about a series like 1/(2^x)?

Answer 6

No, that again is a polynomial term so it stays a finite series.On the other hand,
if you had something like 1/(1+x) that is not of the form of ax^n (where a is some coefficient and n the power to which x is raised). So in such a case we expand it using a binomial expansion of (1+x)^-1 which becomes an infinite series.

Answer 7

The whole purpose of the Taylor series is to represent functions as polynomials. If they are already polynomials then they remain a finite series and no expansion happens.

Answer 8

Hey but I thought 1/(2^x) is not a polynomial? (as opposed to 1/(x^2) I may be wrong...

Answer 9

No you're right, 1/2^x will have a taylor expansion.Sorry, I got confused with 1/x^2.

Answer 10

Thank you :)