Recurrence relations, convergence tests, identifying sequences
3
votes
5answers
47 views
Condition for a common root in two given quadratic equations
If $a,\;b,\;c$ are in Geometric Progression, then the equations $ax^2+2bx+c=0$ and $dx^2+2ex+f=0$ have a common root if $\;\displaystyle\frac da,\;\frac eb,\;\frac fc$ are in:
Arithmetic Progression
...
2
votes
3answers
76 views
Proof $\lim\limits_{n \rightarrow \infty}n(a^{\frac{1}{n}}-1)=\log a$
I want to show that for all $a \in \mathbb{R }$
$$\lim_{n \rightarrow \infty}n(a^{\frac{1}{n}}-1)=\log a$$
So far i've got $\lim\limits_{n \rightarrow \infty}ne^{(\frac{1}{n}\log a)}-n$, but when i ...
3
votes
3answers
70 views
Proving that $\frac{3}{2} \sum_{k=1}^{\infty} \frac{4}{k^3+k^2} = \pi^2-6$
I'm trying to prove that:
$$\frac{3}{2} \sum_{k=1}^{\infty} \frac{4}{k^3+k^2} = \pi^2-6$$
I've tried looking at the partial sums, but no luck there. I just have no idea where to begin.
Knowing that ...
1
vote
3answers
43 views
Counter example regarding alternating series test [duplicate]
Q: Is $\sum_2^\infty{(-1)^n}{a_n}$ convergent?
$$a_n={1\over \sqrt n+(-1)^n} \space \forall n\in\mathbb{N_{\ge2}}$$
The answer says it diverges.The only thing I could deduce is that the series is not ...
1
vote
1answer
42 views
Prove the series identity
Prove an identity:
$$\sum_{n=2}^{ \infty } \frac{2}{(n^3-n)3^n}=- \frac{1}{2}+ \frac{4}{3} \cdot \sum_{n=1}^{ \infty } \frac{1}{n \cdot 3^n}$$
I've checked that the left-hand-side of this ...
1
vote
4answers
26 views
Finding limit of a recursively defined sequence
Let $(x_{n})_{n\geq1}$ be a sequence defined by:
$x_1=1$ and $x_n=n(x_{n+1}-\frac{n+1}{n^2})$. Calculate $\lim_{x\rightarrow \infty }nx_n$.
We can write $x_n=n(x_{n+1}-\frac{n+1}{n^2})$ as $(n+1) ...
-5
votes
1answer
56 views
Which series converge to $2$? [on hold]
Which series converge to $2$?
Infinite series of triangular numbers and $2^n$ converge to $2$. Is there any other which converge to $2$?
@all. I don't understand $a_n$ or $b_n$, talk in simple words. ...
0
votes
2answers
42 views
Value of Riemann zeta function at $-1$
This claim is false $\sum_{n=1}^{\infty}n=\sum_{n=1}^{\infty}n^{-(-1)}= \zeta(-1)=-1/12$.
The error is that we should
$\sum_{n=1}^{\infty}n=\sum_{n=1}^{\infty}(1/n ^1)^{-1}=(0)^{-1}$.
Am I correct? ...
0
votes
1answer
37 views
Is this sequence convergent, divergent or oscillatory?
Consider the sequence $\displaystyle{\left\{n\sin\left(\pi \over n\right)\right\}}$,
$\displaystyle{n = 1, 2, 3,\ldots}$
Is the sequence convergent, divergent or oscillatory? And why?
0
votes
3answers
39 views
Real root of a real equation2
I would appreciate if somebody could help me with the following problem
Q: If $ \alpha_n$ is a real root of equation $x^n=-x+1$ and
$ \alpha_{n+1}$ is a real root of equation $x^{n+1}=-x+1$ then show ...
0
votes
1answer
37 views
Is this sequence using tan, convergent, divergent or oscillatory?
What is the nature of the sequence $\left\{(-1)^n \tan \left( \frac{\pi}{2} - \frac{1}{n} \right )\right\}_{n \geq 1}$?
Is this sequence convergent, divergent or oscillatory?
If it divergent, does it ...
0
votes
1answer
44 views
Fourier Series Approximations of Functions
From a few examples of smooth functions, discontinuous functions and continuous functions which have a 'kink' (i.e. $|x|$ where left and right limits disagree)... I've seen that the fourier series ...
1
vote
1answer
43 views
complex irreps is in bijective correspondence with sequences
Let $\{a_n\}$ be a sequences of positive integers such that
$$0 \leq a_n\leq p^n - 1,$$
$$a_n \equiv a_{n +1} \bmod p^n \quad \text{for all $n$}$$
Prove that the complex irreps of the group $ ...
5
votes
2answers
73 views
Where is $f(x) := \sum_{n=1}^\infty \frac{\langle nx\rangle}{n^2+n}$ discontinuous?
Let $\langle x\rangle$ denote the fractional part of $x\in
\mathbb{R}$, i.e. $\langle x\rangle := \inf \{ x-k: k\in \mathbb{Z},
k\leq x \}$.
Define $$f(x) := \sum_{n=1}^\infty ...
1
vote
1answer
22 views
If every sequence that approaches a point $c$ has a codomain limit of $L$, then the limit of the function is also $L$.
Let $f$ be a fuction, $D$ its domain, and $c$ a point of it. If we have that
$$\forall (x_n)_{n\in\mathbb{N}} \subset D : \lim x_n=c, x_n \neq c \longrightarrow L=\lim f(x_n)$$
That is, all ...
