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Results tagged with real-analysis
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user 1243448
For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.
3
votes
2
answers
71
views
Show that if the sequence $\{a_n\}_{n=1}^\infty$ converges to $a$, then $\lim_{n\to\infty}\f...
Show that if the sequence $\{a_n\}_{n=1}^\infty$ converges to $a$, then $\lim_{n\to\infty}\dfrac{a_1+a_2+\dots+a_n}{n}=a.$
I am not trying to use the Stolz-Cesaro theorem.
What I am actually trying to …
- 113
0
votes
4
answers
58
views
Show that $\lim_{n\to\infty}\frac{n^p}{a^n}=0,p\in\mathbb{N},a>1$ [duplicate]
Show that $$\lim_{n\to\infty}\dfrac{n^p}{a^n}=0,p\in\mathbb{N},a>1$$
I was really impressed when I understood that we can choose any $p$ and $a$ (satisfying $p\in\mathbb{N},a>1$) and the limit still h …
- 113
1
vote
2
answers
58
views
Find the limit $\lim_{n\to\infty}\left(\frac{n^2-4n+3}{n^2-7n+10}\right)^{n+\sin(n!)}$
Find the limit $$\lim_{n\to\infty}\left(\dfrac{n^2-4n+3}{n^2-7n+10}\right)^{n+\sin(n!)}$$
Unfortunately $n^2-4n+3$ (the numerator) and $n^2-7n+10$ (the denominator) don't have a common root (or common …
- 113
1
vote
2
answers
95
views
Show that a function is $f$ bijective if $f(f(f(2x+3)))=x$ for all real $x$
Let $ f : \mathbb{R} \to \mathbb{R} $ is a function such that
$$ \forall x\in\mathbb{R} : f(f(f(2x+3)))=x $$
Show that $f$ is bijective.
We have to show that $f$ is injective and surjective.
How do we …
- 113
0
votes
1
answer
63
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Examine the convergence of a sequence
For $\alpha\in \mathbb{R}$, let $a_1=\alpha$ and $a_{n+1}=2-\dfrac{2}{a_n^2+1}$. Examine the convergence of the sequence ${\{a_n\}}_{n=1}^{\infty}$ for different values of $\alpha$. Also find $\lim_{n …
- 113