1. ### Can someone help me with this Taylor series expansion?

I tried diffrentiating upto certain higher orders but didn’t find any way.. is there a trick or a transformation involved to make this task less hectic? Pls help
2. ### Taylor/Maclaurin series of a function

Homework Statement Obtain Maclaurin Series for: f(x) = sin(x2)/x Homework Equations f(x) = ∑f(n)(c) (x-c)n / n! (for Maclaurin c = 0) The Attempt at a Solution I know that sin(x2) = x2 - (x2*3/3! +... from the final answer I see, that this is just multiplied to 1/x. This bothers me...
3. ### Is this question incomplete? Regarding entire functions...

Homework Statement Let ##F## be an entire function such that ##\exists## positve constants ##c## and ##d## where ##\vert f(z)\vert \leq c+d\vert z\vert^n, \forall z\in \Bbb{C}##. Is this question incomplete? My complex analysis course is not rigorous at all and this came up on a past final...
4. ### Finding a the value of 30th derivative given power series.

Homework Statement The problem is attached as pic Homework Equations ∑(ƒ^(n)(a)(x-a)^n)n! (This is the taylor series formula about point x = 3) The Attempt at a Solution So I realized that we should be looking at either the 30th,31st term of the series to determine the coefficient. After we...
5. ### I Series Expansion to Function

I ran across an infinite sum when looking over a proof, and the sum gets replaced by a function, however I'm not quite sure how. $$\sum_{n=1}^\infty \frac{MK^{n-1}|t-t_0|^n}{n!} = \frac{M}{K}(e^{K(t-t_0)}-1)$$ I get most of the function, I just can't see where the ##-1## comes from. Could...
6. ### Confusing log limit

Homework Statement \lim\limits_{x \to 0} \left(\ln(1+x)\right)^x Homework Equations Maclaurin series: \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} + ... + (-1)^{r+1} \frac{x^r}{r} + ... The Attempt at a Solution We're considering vanishingly small x, so just taking the first term in the...
7. ### Differential equation with power series

Homework Statement Solve y''+(cosx)y=0 with power series (centered at 0) Homework Equations y(x) = Σ anxn The Attempt at a Solution I would just like for someone to check my work: I first computed (cosx)y like this: (cosx)y = (1-x2/2!+x4/4!+ ...)*(a0+a1x+a2x2 +...)...
8. ### Proof Taylor series of (1-x)^(-1/2) converges to function

Hello, I want to prove that the taylor expansion of f(x)={\frac{1}{\sqrt{1-x}}} converges to ƒ for -1<x<1. If I didn't make a mistake the maclaurin series should look like this: Tf(x;0)=1+\sum_{n=1}^\infty{\frac{(2n)!}{(2^n n!)^2}}x^n My attempt is to use the lagrange error bound, which is...
9. ### Conceptual: Are all MacLaurin Series = to their Power Series?

Homework Statement To rephrase the question, given a power series representation for a function, like ex , and its MacLaurin Series, when I expand the two there's no difference between the two, but my question is: Is this true for all functions? Or does the Radius of Convergence have to do with...