Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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Let $a_{2n-1}=-1/\sqrt{n}$ for $n=1,2,\dots$ Show that $\prod (1+a_n)$ converges but that $\sum a_n$ diverges.

Let $a_{2n-1}=-1/\sqrt{n}$, $a_{2n}=1/\sqrt{n}+1/n$ for $n=1,2,\dots$ Show that $\prod (1+a_n)$ converges but that $\sum a_n$ diverges. What I have found so far is that $\prod_{k=2}^{2n} ...
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Finf f such that $F \circ F$ is a primitive of f

Find all primitivable functions $f:\mathbb{R} \to \mathbb{R}$ that admits a primitive $F:\mathbb{R} \to \mathbb{R}$ for which $F\circ F$ is a primitive of $f$. From $(F \circ F)'=f$ we get that ...