### Questions (29)

 6 $T \in \mathcal L (V)$ has no real eigenvalues. Prove that every subspace of $V$ invariant under $T$ has even dimension. 3 Need to compute $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)(2n+2)3^{n+1}}$. Is my solution correct? 3 Solve the differential equation $\sin x\frac {dy}{dx}+(\cos x)y=\sin(x^2)$ [closed] 2 Probability of rolling three different dice [duplicate] 2 Evaluate $\lim_{x\to \infty} (\frac {x+1}{x-1})^x$

### Reputation (353)

 -2 Show that $f(x)=\sum_{n=1}^{\infty}b_n\sin nx$ and $f'(x)=\sum_{n=1}^{\infty}nb_n\cos nx$ coverge uniformly for all x. +10 Prove that$|\int_{D}f(x)dx|\leq\int_{D}|f(x)|dx$. +10 $T \in \mathcal L (V)$ has no real eigenvalues. Prove that every subspace of $V$ invariant under $T$ has even dimension. +10 Find the curvature of the ellipse by the explicit equation $y=\pm\frac{b}{a}\sqrt{a^2-t^2}$

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### Tags (30)

 0 integration × 6 0 exponential-function × 2 0 linear-algebra × 6 0 general-topology × 2 0 limits × 3 0 sequences-and-series × 2 0 geometry × 3 0 taylor-expansion 0 eigenvalues-eigenvectors × 2 0 vector-space-isomorphism

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