[Community Question] Calculus: Limit of a power series

One of our user asked:

The series of functions $$ \sum_{n=0}^{+\infty} \frac{(-x)^n}{1+n!} $$ is pointwise convergent for any $x\in \mathbb{R}$, thus it defines a function $f:\mathbb{R}\rightarrow \mathbb{R}$. Is there a way to evaluate the limit $\lim_{x\rightarrow +\infty} f(x)$?

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Calculus: Intersection of hyperplanes in $\mathbb{R}^5$

Consider the $4$ hyperplanes in $\mathbb{R}^5$ given by the equations $$x_1-x_2+x_3+x_4+2x_5=0$$ $$2x_1-x_2+6x_3+2x_4+6x_5=0$$ $$3x_1-3x_2+3x_3+4x_4+7x_5=0$$ $$x_1+x_2+9x_3+x_4+6x_5=0$$ Let $V \leq \mathbb{R}^5$ be the intersection of these hyperplanes. Find a basis for $V$ .

[Community Question] Calculus: calculating 2 constants in a function

One of our user asked: $$M=\{f\in C[0,2\pi],\int_{0}^{2\pi}f(x)sinxdx=\pi,\int_{0}^{2\pi}f(x)sin2xdx=2\pi\} $$ $a,b\in \mathbb R, g\in M, g(x)=asinx+bsin2x,x\in [0,2\pi]$ I've read on the answers that $a=1,b=2$ and I don't know how to calculate them. Can somebody explain me,please? By the way, the problem is to determine $ \int_{0}^{2\pi}(g(x))^2 dx$ so you have to first get the constants $a$ and $b$ .

Calculus: A difficulty in understanding a step in a solution.

Here is the solution: But I could not understand how the last term in the fourth line came from the line before it, could anyone explain this for me please?

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