# 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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### How to evaluate $\sum_{n=1}^{\infty}a_n$?

If $$a_{n}=1-\frac{1}{2}+\frac{1}{3}-\cdots +\frac{\left ( -1 \right )^{n-1}}{n}-\ln 2$$ then how to eveluate $$\sum_{n=1}^{\infty}a_n$$ does it converge?
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### Convergence and sum of series with exponents

So the question is how can I see if this series : $$\displaystyle\sum_{n=1}^{\infty} \frac{1}{(4-(-1)^n)^n}$$ converges and find its sum. So I would probably need to use the Leibnitz criterion for ...
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### Does this alternating sum of roots converge to $\sqrt2$?

This problem arose from what I'm hesitant to call an investigation into a certain type of "quadrature". Starting with the unit disk, I partition it into $p$ pieces by cutting the disk with vertical ...
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### Three negative and three positive $1$ s in a serie(updated)

I asked the same question here:Three negative and three positive $1$s in a serie But when I see that it was closed because of being unclear I decided to make It better and ask again. First I want to ...
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### If $a_n$ converges to $A$ and a is a non-empty set, show that $4A^2=A^2+4+4/(A^2)$ [on hold]

Consider the example of the sequence defined recursively by $$a_{n+1} = \frac12\left(a_n+\frac{2}{a_n}\right)$$ for $n\geq1$. By easy algebra,$$4a^2_{n+1} = a^2_n+4+\frac{4}{a^2_n}$$ Suppose ...
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### Prove that $\sum_{n=1}^{\infty }\frac{B_{2n}}{(2n-1)!}=\frac{1}{2}-\frac{1}{(e-1)^2}$

Prove that $$\sum_{n=1}^{\infty }\frac{B_{2n}}{(2n-1)!}=\frac{1}{2}-\frac{1}{(e-1)^2}$$ My idea is to find the Taylor series of $\frac{1}{(e^x-1)^2}$, but it seems not useful. Any helps, thanks
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### Which function of $x$, other than $x +c$, and Integral of ($\cos x)^2+(\sin x)^2$ have derivative =$1$

It is a simple question:Which function of x, other than x +c, and Integral of (cosx)^2+/(sinx)^2 have derivative =1. Alternative question, Which other equation gives its derivative as a real number. ...
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### Finding the general formula of a sequence: $3,8,23,68,203,608,\cdots$

I have the following sequence : $$3,8,23,68,203,608,\cdots$$ I have found that definition by recurrence of this is $$a(n)=3a(n-1)-1$$ where $a_0=3$ as the first term. I want to find the explicit ...
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### Convergence of a series of complex numbers.

Let $f : \mathbb C \to \mathbb C$ be a non constant entire function. Does the series $\sum_{n=1}^ \infty \frac{1}{n} f(\frac {z}{n})$ converges at any point $z \in \mathbb C$ ? I think this will not ...