Sunday, 16 November 2014

real analysis - Show that if all convergent subsequences of a bounded sequence converge to $l$, the sequence itself must also converge to $l$.

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Posted here for proof verification and corrections and tips! Let $(a_n){^\infty_{n=0}}$ be a bounded sequence with the property that there e...

calculus - Uniform Continuity and Differentiation

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Is the following true or false?: Let $f\colon [0,1) \to \mathbb{R}$ be a function differentiable in $[0,1)$ (where the derivative at zero...

summation - Prove a lower bound for $sum_{i=1}^n i^2$

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Prove that $$\sum_{i=1}^n i^2 \geq \frac{n^3}{3}$$ for all $n \geq 1.$ What I know: I know the basic format of how to make a proof with th...

sequences and series - Another Simple convergence theorem proof

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Question: Let $\{a_n\}$ and $\{b_n\}$ be convergent sequences with $a_n \Rightarrow L$ and $b_n \Rightarrow M$ as $n \Rightarrow \infty$. Pr...
Saturday, 15 November 2014

sequences and series - Choosing convenient limits of integration on Basel problem

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I have recently discovered $$\sum_{k=1}^{\infty}\frac{\cos\left(k\alpha\right)}{k^{2}}-\sum_{k=1}^{\infty}\frac{\cos\left(k\beta\right)}{k^{...

sequences and series - Problem about sum of arithemtic progression and geometric progression

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Question: An arithmetic sequence has a common difference of $1$ and a geometric sequence has a common ratio of $3$. A new sequence is forme...

calculus - Prove the relation involving derivative of inverse of a function

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I want to prove the following result: $$ {f^{-1}}'(x)=\frac{1}{f'({f^{-1}}(x))}$$ Is simple application of chain rule a valid proof ...