# Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.

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### I have written a maths “paper” about a nice trick that I discovered. Does anyone mind reading it and giving me feedback.

file:///C:/Users/attar/OneDrive/Desktop/The%20Hidden%20Sequence.pdf This is the link to the pdf. To put it into some context: I am a 16 year old mathematics student who is passionate about the subject....
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### $\sum\left(\frac{ n^2 + 1}{n^2 +n + 1}\right)^{n^2}$ converges or diverges?

The original question is to show that $\;\sum\left(\dfrac{ n^2 + 1}{n^2 +n + 1}\right)^{n^2}$ either converges or diverges. I know it diverges but I'm having difficulty arriving at something useful ...
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### Infinity minus a divergent series

I was testing out Wolfram Alpha’s language to math feature and input this query: My understanding is that this would be undefined, as the series diverges; however wolfram states the answer is ...
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### Using formal definition of a limit to prove (-1/2)^n converges to L = 0

Hey there I'm trying to solve a convergence question that uses the formal definition of convergence to show that (-1/2)^n converges to L = 0. I have gotten up to the below stages but I am unsure if it ...
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### Proving two polynomials converge to the same function

I am looking to create polynomials that converge past the usual radius of convergence on the real line. So far, I have proved that this polynomial will converge to the analytic continuation of the ...
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### How can I prove that $\sum_{n=1}^\infty \dfrac{n^4}{2^n} = 150$?

I can easily prove that this series converges, but I can't imagine a way to prove the statement above. I tried with the techniques for finding the sum of $\sum_{n=1}^\infty \frac{n}{2^n}$, but I didn'...
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### Taylor Series representation of $f(x) = \sqrt{x} + \frac{1}{\sqrt x}$ at $a=1$

I am trying to find the Taylor Series representation of $f(x)= \sqrt x + \frac1{\sqrt x}$ at $a = 1.$ With $5$ terms. I know how to get the series expansion. centered at $a=1$. with $5$ terms… However ...
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### Equivalent for $v_{n+1}=v_n\ln(v_n)$

Do you know how to get an equivalent for $(v_n)$ defined by $v_0>\mathrm{e}$ and $\forall n\in\mathbb{N},\; v_{n+1}=v_n\ln(v_n)$ ? Thank you.
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### Prove that the sequence $S_n=\sum_{i=n}^{\infty} x_i$ is cauchy?

Let the sequence $(x_n)_{n\in \mathbb{N}}$ be defined by $lim_{n\to \infty} x_n =0$ and $(x_n)_{n\in \mathbb{N}}$ is monotonically increasing. Prove that $S_n=\sum_{i=n}^{\infty} |x_i|$ is cauchy?
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### If $a_n \neq 0$ for all natural numbers n , $\sum_{n=1}^{+\infty} a_n$ converges $\lim_{n\to+\infty} a_n/b_n =1\sum_{n=1}^{+\infty} b_n$ converges

If $a_n \neq 0$ for all natural numbers $n$ , $\sum_{n=1}^{+\infty} a_n$ converges $\lim_{n\to+\infty} \frac{a_n}{b_n} =1$ then $\sum_{n=1}^{+\infty} b_n$ converges I am trying to find ...
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### Prove that if $\sqrt[n]{x_n} \to \ell$ then $x_{n+1}/x_n \to \ell$

I am stuck with the following problem: Let $\lbrace x_n \rbrace$ be a sequence of positive real numbers. Prove that if \begin{align} \lim_{n\to \infty} \sqrt[n]{x_n} = \ell \Longrightarrow \lim_{n\...
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### Argue that the sequence converges

Let $$a_n=n^2\cos(1/n)-n^2$$ Show that the sequence converges. Now, I know how to use the formal definition of convergence but I am looking for simpler methods (i.e the tests for series). I found it ...
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### Compute $\int_{0}^{1}\frac{\ln(1+x)\ln(1+x^2)}{x}\,dx=\frac{\pi}{2}G-\frac{33}{32}\zeta(3)$

I saw the following result and I am trying to prove it. $G$ is Catalan´s constant. $$\boxed{\int_{0}^{1}\frac{\ln(1+x)\ln(1+x^2)}{x}\,dx=\frac{\pi}{2}G-\frac{33}{32}\zeta(3)}$$ I could not figure out ...