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roman
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27 votes
3 answers
3k views

Reverse engineering a math magic trick involving matrices.

11 votes
3 answers
253 views

Show that $\forall n \in\Bbb N: e < \left(1+{1\over n}\right)^n \left(1 + {1\over 2n}\right)$

11 votes
6 answers
334 views

Proof verification for $\lim_{n\to\infty}\frac{1}{n}(1+\sqrt2+\dots + \sqrt{n}) = +\infty$

11 votes
2 answers
237 views

Study continuity of $f(x) = |x|$ for $x\in\Bbb R \setminus \Bbb Q$, $f(x) = \frac{qx}{q+1}$ for $x\in\Bbb Q$.

8 votes
3 answers
129 views

Show that $x_n = \sqrt{\sum_{k=1}^n\left(\frac{k+1}{k}\right)^2} - \sqrt n$ is a bounded sequence.

7 votes
3 answers
185 views

Show that $x_{n+2} = x_{n+1} + \frac{x_n}{2^n}$ is a bounded sequence

7 votes
6 answers
815 views

A more rigorous way to show that $x^5 - 3x = 1$ has at least $3$ roots in $\Bbb R$

7 votes
1 answer
542 views

Study convergence of $x_{n+1} = x_n^2 + 3x_n + 1$, where $x_1 = a$, and $a$ takes different values and find its limit.

6 votes
3 answers
128 views

Simplifying the result of integration of $\int\frac{e^x + e^{3x}}{1-e^{2x}+e^{4x}}\mathop{dx}$

5 votes
2 answers
855 views

Riemann sum of $\int_1^2 {1\over x^2} dx$.

5 votes
2 answers
112 views

Find all possible values of $a, p$ for which $x_{n+1} = x_n^2 + (1-2p)x_n + p^2$ converges, given $x_1 = a$ and $n\in\Bbb N$

5 votes
2 answers
298 views

Evaluate the limit: $\lim\limits_{n\to\infty} \frac{4^nn!}{(3n)^n}$

5 votes
6 answers
146 views

Elementary way to evaluate $\lim_{x\to0}\frac{\sqrt[n]{a+x} - \sqrt[n]{a-x}}{x}$

5 votes
1 answer
258 views

Prove that $\limsup_{x\to\infty}\left(\cos x + \sin\left(\sqrt2 x\right)\right) = 2$

5 votes
1 answer
203 views

Show that $S_n=1+{x\over1!}+{x^2\over2!}+\cdots+{x^n\over n!}$ converges for $n\in\Bbb N,\ x \in\Bbb R$ without using Taylor series.

4 votes
3 answers
106 views

Can't find a seemingly simple limit $\lim_{n\to\infty}\frac{(n+k)!}{n^n}$

4 votes
5 answers
193 views

Verify the proof that $x_n = \ln^2(n+1) - \ln^2n$ is a bounded sequence.

4 votes
2 answers
20k views

Detect double error using Hamming code.

4 votes
1 answer
454 views

How to compute $k$'th roots $\!\bmod m$?

4 votes
5 answers
285 views

Prove that if ${x_1, x_2, x_3}$ are roots of ${x^3 + px + q = 0}$ then ${x_1^3+x_2^3 + x_3^3 = 3x_1x_2x_3}$

4 votes
1 answer
364 views

Check the proof that $f(x) = \cos({x})\cdot\cos({\sqrt{3}x})$ is not periodic

4 votes
1 answer
2k views

Prove a sequence with bounded variation converges.

4 votes
4 answers
241 views

Show that $\lim_{n}\sum_{k=n}^{2n}{1\over k} = \ln2$ using elementary methods.

4 votes
1 answer
189 views

$\{x_n\}$ is a bounded above sequence such that $x_{n+1} - x_n \ge a_n$, where $\sum a_k$ converges. Prove $x_n$ converges.

4 votes
1 answer
81 views

Verification of the following integral $\int e^{-x}\arctan e^x\ dx$.

4 votes
4 answers
136 views

Explanation of an integration trick for $\int \frac{a_1 \cos x + b_1 \sin x}{a\cos x + b\sin x}dx$

3 votes
1 answer
159 views

Evaluate $\int {dx\over \sqrt[6]{(x-7)^7(x-5)^5}}$

3 votes
1 answer
3k views

Reduction formula for $\int \cosh^n x dx$

3 votes
1 answer
551 views

Alternative proof of reduction formula for $\int {dx \over (a\sin x + b\cos x)^n }$

3 votes
1 answer
210 views

Prove a specific piecewise function is continuous.

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