Intereting Posts

Pseudocompactness in the $m$-topology in $C(X)$
Values for $(1+i)^{2/3}$
Prob 12, Sec 26 in Munkres' TOPOLOGY, 2nd ed: How to show that the domain of a perfect map is compact if its range is compact?
Integrable – martingale
$x^p-x-1$ is irreducible over $\mathbb{Q}$
Poincare-Bendixson Theorem
Proving the integral converges for all $p>1, q<1$
Thickness of the Boundary Layer
How to prove this inequality about the arc-lenght of convex functions?
Can we define probability of an event involving an infinite number of random variables?
Ramsey Number proof: $R(3,3,3,3)\leq 4(R(3,3,3)-1) + 2$
Dirichlet's theorem on primes in arithmetic progression
Intermediate field between $F$ and $F(x)$
prove $\sum_{i=0}^{n}\binom{2n+1}{i}=2^{2n}$
Wolfram Alpha can't solve this integral analytically

A question in a past paper says prove that this series converges pointwise but not uniformly $$\xi(x):= \sum_{n=1}^\infty \frac{1}{n^x} .$$ But I thought that it did converge uniformly to some function $\xi(x):(1,\infty) \to \mathbb{R}$. Here’s why; if you work on $(1+\delta,\infty)$ for $\delta >0$, then clearly we have $$\left\|\frac{1}{n^x}\right\|_\infty = \frac{1}{n^{1+\delta}}$$ and clearly $\sum \frac{1}{n^{1+\delta}}$ converges so by the Weierstrass M-test $\sum_{n=1}^\infty \frac{1}{n^x}$ converges uniformly on $(1+\delta,\infty)$ so letting $\delta \to 0$ gives $\sum_{n=1}^\infty \frac{1}{n^x}$ converging uniformly on $(1,\infty)$?

- Is this a metric on R?
- Cluster Point in Fréchet-Urysohn Space is Limit Point?
- Dirichlet series
- Uniform convergence of geometric series
- Calculating in closed form $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^4(m^2+n^2)}$
- How compute $\lim_{p\rightarrow 0} \|f\|_p$ in a probability space?
- Measurability of limit function into a Polish space
- Prove that the following set is dense
- The completion of a separable inner product space is a separable Hilbert space
- Differentiation under integral sign (Gamma function)

**Hint:** Use an estimate that gives *concrete* information about the error when you cut off so that the first term left out involves $\frac{1}{m^x}$. This error is $\sum_m^\infty \frac{1}{n^x}$, which is greater than

$$\int_{m}^\infty\frac{dt}{t^x}.$$

The above integral is equal to

$$\frac{1}{x-1}\frac{1}{m^{x-1}}.\tag{$\ast$}$$

Now show that however large $m$ may be, there is an $x$ such that the expression in $(\ast)$ is well away from $0$. You might for example use $x=1+\frac{1}{m}$.

- Number of partial functions between two sets
- What does “IR” mean in linear algebra?
- Analytic solution to the one-compartment model
- Does Hartshorne *really* not define things like the composition or restriction of morphisms of schemes?
- Conditional probability containing two random variables
- Eigenvalue problems for matrices over finite fields
- Upper triangular matrix and nilpotent
- In which of following stuctures is valid implication $x\cdot y=1\implies x=1$?
- Boundedness and Cauchy Sequence: Is a bounded sequence such that $\lim(a_{n+1}-a_n)=0$ necessarily Cauchy?
- Elementary properties of gradient systems
- How can I prove $\lim_{n \to \infty} \int_{0}^{\pi/2} f(x) \sin ((2n+1) x) dx =0 $?
- Why does this covariance matrix have additional symmetry along the anti-diagonals?
- If $R$ and $R]$ are isomorphic, then are they isomorphic to $R$ as well?
- Discriminant of splitting field
- Understanding the symmetries of the Riemann tensor