# Tagged Questions

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### Taylor series expansion for $e^{\sin{x}}$

Given the function $$f(x)= e^{\sin{x}}$$ I have to write it without using the exponential or sine function. I came to this point $$f(x) = \sum_{k=0}^{\infty} \frac{\sin^k{x}}{k!}$$ How can I get ...
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### Use Taylor's Theorem and Taylor Expansion to prove [on hold]

Suppose $f: R\rightarrow R$ is such that both $f'$ and $f''$ exist for all $x \in \mathbb{R}$, so that Taylor's Theorem tells us that, $\forall a, h \in \mathbb{R}, \exists θ$ such that $0<θ<1$ ...
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### Lagrange Remainder and Intervals of convergence

(a) Determine the largest interval centered at $c=0$ on which we can be sure that $\lvert \cos(x) -(1-\frac{x^2}{2})\rvert < 10^{-6}$ (b) Let $T_n(x)$ denote the Taylor polynomial of order $n$ for ...
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### What is series coefficient for $f(x)=\csc^2 x - \frac1{x^2}$?

What is general formula for Maclauren series expansion for $f(x)=\csc^2 x - \frac1{x^2}$ ?
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### Infinitely differentiable function with divergent Taylor series?

I'd greatly appreciate it if someone could provide examples of the following: 1) A infinitely differentiable function whose Taylor series does not converge to the function. 2) An infinitely ...
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### If it converges, how to show that power series converges to $f(x)$?

I had a very basic question. Suppose $f(x)$ is a function. And let us say it has a power series :- $$f(x) = \sum_{n=0}^\infty a_nx^n.$$ Suppose we are operating inside the region of convergence. ...