19 Does the sequence $\{\sin^n(n)\}$ converge? 9 $p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x)$, then all the roots of $p_k(x)$ are real 6 $f(x+1)-f(x)$ converges $\Rightarrow\frac{f(x)}x$ converges 6 convergente series $\sum_{n=1}^{+\infty}a_n$ such that all the power diverges 5 Find $f$ such as $f(x) = \sum_{n=1}^\infty \frac{f(x^n)}{2^n}$

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### Tags (26)

 45 sequences-and-series × 10 10 polynomials × 2 39 real-analysis × 5 9 real-numbers 27 convergence-divergence × 4 9 roots 19 diophantine-approximation 5 summation × 2 11 functions × 2 4 taylor-expansion

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 282 What is the importance of eigenvalues/eigenvectors? 261 How does one prove the determinant inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$? 200 Any open subset of $\Bbb R$ is a countable union of disjoint open intervals 180 A Topology such that the continuous functions are exactly the polynomials 166 How to define a bijection between $(0,1)$ and $(0,1]$?