# Tagged Questions

Questions regarding the Taylor series expansion of univariate and multivariate functions, including coefficients and bounds on remainders. A special case is also known as the Maclaurin series.

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### How to show $1 +x + x^2/2! + \dots+ x^{2n}/(2n)!$ is positive for $x\in\Bbb{R}$?

How to show $1 + x + \frac{x^2}{2!} + \dots+ \frac{x^{2n}}{(2n)!}$ is positive for $x\in\Bbb{R}$? I realize that it's a part of the Taylor Series expansion of $e^x$ but can't proceed with this ...
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### Taylor series of $\ln(x+2)$

I try to determine the Taylor series of $\ln(x+2)$. Since I know $\ln(1-x) = \sum_{n=1}^{\infty} \frac{x^n}{n}$, I suppose I can rewrite, \begin{align} \ln(x+2) &= \ln(1-(-(x+1)))=-\sum_{n=1}^{\...
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### Taylor series and radius of convergence: $\sqrt{x}$ with centre $x = 16$?

I've been struggling with this question for a while now and getting nowhere with it. Could someone please help me out? Assuming that the function has a power series expansion about the given point, ...
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### Taylor Series and Differentiation with Sigma notation $f(x) = \frac{x}{(2-3x)^2}$

Use Term By Term Differentiation to Find the Taylor Series about $x$=3 for Give The Open Interval of Convergence and express as sigma notation $\sum A_n(x-3)^n$ $f(x) = \frac{x}{(2-3x)^2}$ So I ...
I'm trying to evaluate the limit as $N\to \infty.$ $$\frac{ \left(\dfrac{\sin \frac{1}{N}} {\frac{1}{N}}\right)^{N} -1 }{\frac{1}{N}}.$$ Note first that, using L'Hôpital, one can easily show that ...
### Find smallest $k$ such that the given trigonometric functions are $O(x^k)$
I feel like I do not quite grasp the concept of Big O Notation. From my understanding, if $f(x) = O(g(x))$ then $f(x)$ is at most $g(x)$ multiplied by some constant C, which makes decent sense to me. ...