# 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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### Series Relation

$\{A_n\}$ is a sequence of 4th order tensors. $lim_{n\rightarrow\infty}A_n = O_4$, where $O_4$ is the null 4th order tensor. The series $\sum_{n=1}^{\infty}A_n$ converge to a known tensor $B$. I ...
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### Summation of a series involving the MOD function

Evaluate the sum of: $$\sum_{n=0}^{\infty} (n\bmod 3)\cdot 2^{-n}$$ Any idea how can this sum be evaluated?
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### help in real analysis [on hold]

How can I use this definition to prove that $a^{\frac{1}{n}}$ converge to $1$? where a >0
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### Approximation of series using integral

In notes of statistical physics I found the following approximation $$\sum\limits_{n=0}^{\infty}F\left(n+\frac{1}{2}\right)\approx \int_{0}^{\infty}F(x)dx+\frac{1}{24}F'(0)$$ for $F$ such that the ...
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### Boundedness and convergence of $x_{n+1} = x_n ^2-x_n +1$

Suppose that $x_0 = \alpha \in \mathbb{R}$ and $x_{n+1} = x_n ^2-x_n +1$. I am asked to study the boundedness of $(x_n)$ and then asked if $(x_n)$ converges. How can I show that $(x_n)$ is bounded? ...
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### Compute $1 \cdot \frac {1}{2} + 2 \cdot \frac {1}{4} + 3 \cdot \frac {1}{8} + \cdots + n \cdot \frac {1}{2^n} + \cdots$ [duplicate]

I have tried to compute the first few terms to try to find a pattern but I got $$\frac{1}{2}+\frac{1}{2}+\frac{3}{8}+\frac{4}{16}+\frac{5}{32}+\frac{6}{64}$$ but I still don't see any obvious ...
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### Will the expression $\sum_{i=1}^{n}{\frac{i^{2}}{n^{2}}}$ converge as n approches infinity?

I have the following expression: $$\lim_{n \to\infty}\ \sum_{i=1}^{n}{(\frac{i}{n})^{2}}$$ I am not quite sure whether it will converge or diverge. Can somebody tell me how to figure it out?
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### Generalization of Mill's theorem

Mill's theorem states that there exists a positive real number A such that the floor of the double exponential function $A^{3^n}$ are primes for all positive integers n. The value of A is ...
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### Challenging Series example [duplicate]

Let $\{x_n\}$ be a decreasing sequence such that the series of $x_n$ converge. Show that the limit as $n$ approaches infinity of $\{nx_n\}$ equals zero.
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### how is it that $\int_0^x 2\pi r\ dr$ is equal to the area of a circle [on hold]

I'm studying calculus and I'm having some basic questions, this one is regarding the area of a circle. we know, from some guy, that the circumference of a circle is $2 \pi r$ and the area can be seen ...
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### I need help understanding The Integral Test for series

For the following Series I have to show that the series qualifies for The Integral Test, then use it to determine if the series converges or diverges. here's my work where I apply the Integral Test, ...
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### How do you work out the product of this sequence?

I got a question in my maths paper and I didn't know how to answer it. This was the question: What is: $(1+\frac{1}{2}) (1+\frac{1}{3}) (1+\frac{1}{4})$... All the way up to 98 factors a) What is ...
### Where were my mistakes when I've combined $\eta(s)=\left(1-2^{1-s}\right)\zeta(s)$ and the Fourier series for the fractional part?
Let $s=x+it$ the complex variable, thus we are denoting $\Re s=x$. Combining the identity $$\eta(s)=\left(1-2^{1-s}\right)\zeta(s),$$ that holds for $0<x<1$, where $\zeta(s)$ is the Riemann Zeta ...