Intereting Posts

Nonexistence of the limit $\lim_{(x,y)\rightarrow (0,0)} \frac{x^2y^2}{x^3+y^3}$
Shape operator and principal curvature
$B$ is a Borel set, implies $f(B)$ is a Borel set.
is the hilbert polynomial integer-valued everywhere?
Lipschitz continuity implies differentiability almost everywhere.
Volumes of cones, spheres, and cylinders
How did they simplify this expression involving roots of unity?
Original works of great mathematicians
Finding equation of an ellipsoid
What is a point?
Prove $1^2+2^2+\cdots+n^2 = {n+1\choose2}+2{n+1\choose3}$
Proving a basis for inner product space V when $||e_j-v_j||< \frac{1}{\sqrt{n}}$.
Calculating the order of an element in group theory
Natural derivation of the complex exponential function?
Product of sums of square is a sum of squares.

Possible Duplicate:

Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$

Is the sum of sin(n)/n convergent or divergent?

Give a demonstration that the following series converge:

$$\sum_{n=1}^\infty\frac{\sin(n)}{n}$$

- A continuously differentiable function with vanishing determinant is non-injective?
- Evaluating $\int_{-3}^{3}\frac{x^8}{1+e^{2x}}dx$
- Is the function $e^{-|x|^k}$ analytic on any interval not containing zero?
- Prove that F $ \in \mathbb{R} $ is closed if and only if every Cauchy sequences contained in F has a limit that is also an element of F.
- Simple question: the double supremum
- Limit of an $n$-th Root Proof

In the demonstration we can use only elementary convergence test, for example the Leibniz’s test, condensation test, absolute convergence, ecc… ( these test are known by every student of the first course of analysis 1 )

Thanks!!!

- If $f(0) = f(1)=0$ and $|f'' | \leq 1$ on $$, then $|f'(1/2)|\le 1/4$
- Is $f(x)=\sin(x^2)$ periodic?
- Dimension for a closed subspace of $C$.
- Prove Borel Measurable Set A with the following property has measure 0.
- Prove that the tangent space has the same dimension as the manifold
- $f:B\subset \mathbb{R}^m\longrightarrow \mathbb{R}$ is integrable if and only if its graph has zero volume
- For every irrational $\alpha$, the set $\{a+b\alpha: a,b\in \mathbb{Z}\}$ is dense in $\mathbb R$
- Verification of extension result for Lipschitz functions
- A subspace $X$ is closed iff $X =( X^\perp)^\perp$
- Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.8, Problem 3: What is the norm of this functional?

You can conclude it based on Abel partial summation (The result is termed as generalized alternating test or Dirichlet test). We will prove the generalized statement first.

Consider the sum $S_N = \displaystyle \sum_{n=1}^N a(n)b(n)$. Let $A(n) = \displaystyle \sum_{n=1}^N a(n)$. If $b(n) \downarrow 0$ and $A(n)$ is bounded, then the series $\displaystyle \sum_{n=1}^{\infty} a(n)b(n)$ converges.

First note that from Abel summation, we have that

$$\sum_{n=1}^N a(n) b(n) = \sum_{n=1}^N b(n)(A(n)-A(n-1)) = \sum_{n=1}^{N} b(n) A(n) – \sum_{n=1}^N b(n)A(n-1)\\

= \sum_{n=1}^{N} b(n) A(n) – \sum_{n=0}^{N-1}^N b(n+1)A(n) = b(N) A(N) – b(1)A(0) + \sum_{n=1}^{N-1} A(n) (b(n)-b(n+1))$$

Now if $A(n)$ is bounded i.e. $\vert A(n) \vert \leq M$ and $b(n)$ is decreasing, then we have that

$$\sum_{n=1}^{N-1} \left \vert A(n) \right \vert (b(n)-b(n+1)) \leq \sum_{n=1}^{N-1} M (b(n)-b(n+1))\\ = M (b(1) – b(N)) \leq Mb(1)$$

Hence, we have that $\displaystyle \sum_{n=1}^{N-1} \left \vert A(n) \right \vert (b(n)-b(n+1))$ converges and hence $$\displaystyle \sum_{n=1}^{N-1} A(n) (b(n)-b(n+1))$$ converges absolutely. Now since

$$\sum_{n=1}^N a(n) b(n) = b(N) A(N) + \sum_{n=1}^{N-1} A(n) (b(n)-b(n+1))$$

we have that $\displaystyle \sum_{n=1}^N a(n)b(n)$ converges. In your case, $a(n) = \sin(n)$. Hence, $$A(N) = \displaystyle \sum_{n=1}^N a(n) = \dfrac{\sin((N+1)/2) \sin(N/2)}{\sin(1/2)} \leq \csc(1/2)$$ is bounded. Also, $b(n) = \dfrac1n$ is a monotone decreasing sequence converging to $0$.

Hence, we have that $$\sum_{n=1}^N \dfrac{\sin(n)}n$$ converges.

- Derivative of sinc function
- Dimension of spheres in sphere bundles
- Derivative (or differential) of symmetric square root of a matrix
- Why does volume go to zero?
- Find $\displaystyle\lim_{n\to\infty}\sqrt{n}(\sqrt{n + a} – \sqrt{n}), a > 0$
- How to proof that a finite-dimensional linear subspace is a closed set
- An example of a non Noetherian UFD
- Show that the ideal $ (2, 1 + \sqrt{-7} ) $ in $ \mathbb{Z} $ is not principal
- Understanding the graph for $x^y = y^x$
- An incorrect answer for an integral
- Integrating absolute value function
- Is $ 0.112123123412345123456\dots $ algebraic or transcendental?
- Is there a complex variant of Möbius' function?
- difference between linear transformation and its matrix representation .
- If $K^{\mathrm{Gal}}/F$ is obtained by adjoining $n$th roots, must $K/F$ be as well?