# Tagged Questions

For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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### Uniform convergence of $x\arctan(nx)$

Let $f_n(x)=x\arctan(nx)$ for $n\geq1$. Show that $(f_n(x))$ converges uniformly to a function $f$. In the previous parts I have proved that $f_n(x)$ continuously differentiable and it converges ...
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### What are the next few “tetranacci-like” pseudoprimes?

Starting with $n=0$: $k=2$ Given the roots $x_i$ of $x^2-x-1=0$. Then, we have the Lucas numbers, $$A_n = x_1^n+x_2^n = 2, 1, 3, 4, 7, 11, 18,\dots$$ The $n$ that divides $A_n-1$ are all the ...
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### For which $z \in \Bbb{C}$ does this series converge?

How can I determine for which $z\in \Bbb{C}$ the following series converges? $$\sum_{n=0}^{\infty} \frac{z^n}{1+z^{2n}}$$ I've tried the root and ratio test with no success.
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### How to find this sum: $\sum_{n=1}^{\infty}\frac{H^{(r)}(n)}{2^n n}$

I am learning new sums and I would appreciate your hints about how to approach the following $$S(r)=\sum_{n=1}^{\infty}\frac{H^{(r)}(n)}{2^n n}$$ where $$H^{(r)}(n)=\sum_{j=1}^{n}\frac{1}{j^r}$$ is ...
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### Verification of Solution for Walter Rudin Principles of Mathematical Analysis Exercise 20, Chapter 3

I have written an answer for the problem 20, chapter 3 of Walter Rudin's Principle of Mathematical Analysis. I think the proof is correct, but since I am new with this kind of proofs, I am skeptical ...
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### how to integrate this equation with two sums inside?

i'm reading a book, and i have trouble with this problem, i don't how to integrate this equation and where to begin from. Although they already give the answer but I don't understand how to get it. I ...
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### prove or disprove this theorem about subsequences

I need to prove or disprove that for a certain sequence $a_n$, if the subsequence of the even indices and the subsequence of the odd indices are Cauchy sequences then $a_n$ converges.
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### Differentiation method for evaluating $\sum_{n=1}^\infty \frac{n^2}{3^n}$

I evaluated the following infinite sum (the original and broader question regarding this sum can be found at Evaluating $\sum_{n=1}^\infty \frac{n^2}{3^n}$). $$\sum_{n=1}^\infty \frac{n^2}{3^n}$$ ...
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### Find x in the following equation: [closed]

Find x in the following equation: $$\frac{x}{\sqrt{2}+1}+\frac{x}{\sqrt{3}+\sqrt{2}} +\frac{x}{\sqrt{4}+\sqrt{3}}+...+\frac{x}{\sqrt{2025}+\sqrt{2024}}=4004$$
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### Given a summation figure out the alternating series

I figured out that the top is (2x-1) and that the difference between the denominator ends up being (2x-1), just not sure how to figure out what the series is.
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### Prove that $a_{n+1}=a_{n} + 1/(n+1)^2$ is bounded from above

Prove that $a_{n+1}=a_{n} + 1/(n+1)^2$ is bounded from above What I have tried is $$a_n=1+1/4+\dots+1/n^2\leq 1+1+\dots +1=n$$ So I conclude that $a_n$ is bounded above by $n$. Does this ...
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### If $x_n$ $\rightarrow$ 1 Then show that sequence $\frac {4+ (x_n)^{2}}{x_n}$ approaches to limit 5

If $x_n$ $\rightarrow$ 1 Then show that sequence $\frac {4+ (x_n)^{2}}{x_n}$ approaches to limit 5 I have tried to find epsilon proof ,But i am not successful .Can anyone help me with this ...
### Prove that $(n+\frac {(-1)^n}{n})$ is not Cauchy
Let $x_n=(n+\dfrac {(-1)^n}{n})$ So, $|x_{2m}-x_{2n}|=|(2m+\dfrac {(-1)^{2m}}{2m})-(2n+\dfrac {(-1)^{2n}}{2n})|$ $=|2m-2n+\dfrac{1}{2m}-\dfrac{1}{2n}|$ $=|(m-n) (2-\dfrac{1}{2mn})|$ This is ...