# 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 of Functions and Continuity

Let $a > 0$, and $(f_n)_{n=0}^{\infty}$ a sequence of continuous functions $f_n:[-a,a] \rightarrow \mathbb{R}$. Assume that the series $$\sum_{n=0}^{\infty} x^n f_n(t)$$ ...
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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 ...
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### How to solve $x_{100}$ with $x_{n+1}=\frac{2x_n}{2+x_n}+1$ and $x_{1}=3$

How to solve $x_{100}$ with $x_{n+1}=\frac{2x_n}{2+x_n}+1$ and $x_{1}=3$? Can anybody shed light on this? regards.
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### 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 ...
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### How to find the n'th number in this sequence

The first numbers of the sequence are ...
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### Complex Analysis Solution to the Basel Problem ($\sum_{k=1}^\infty \frac{1}{k^2}$)

Most of us are aware of the famous "Basel Problem": $$\sum_{k=1}^\infty \frac{1}{k^2} = \frac{\pi^2}{6}$$ I remember reading an elegant proof for this using complex numbers to help find the value of ...
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### How to prove $\ln{6}=\sum_{n=1}^{\infty}\sum_{r=2}^{\infty}\left({1\over r^{2n}}+{2\over (r+1)^{2n}}+{1\over (r+2)^{2n}}\right)$?

I need help, on how to prove $$\ln{6}=\sum_{n=1}^{\infty}\sum_{r=2}^{\infty}\left({1\over r^{2n}}+{2\over (r+1)^{2n}}+{1\over (r+2)^{2n}}\right).$$ Any hints?
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### Proving a necessary and sufficient condition for compactness of a subset of $\ell^p$

Let $A \subset \ell^p$, where $1 \le p \lt \infty$. Suppose the following conditions are true: 1) $A$ is closed and bounded 2) $\forall \epsilon \gt 0, \: \exists \: N \in \mathbb{N}$ such ...
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### Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
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### Proof related to the least squares method

I've seen this exercise in several statistics text, but how they get to the final formula is something that I don't quite get. How do two squared terms suddenly become a binomial term? I've been ...
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### Method for writing set intersections/unions/complements/etc. in terms of polynomial functions? [closed]

I realize that I'm woefully uninformed regarding most math concepts, and I also realize (given the aforementioned fact) that I'm probably not using the correct format/notation, so please, be gentle. ...
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### What is the sum of $\sum_{i=0}^\infty \frac{i^n}{4^i}$ in terms of n? [closed]

$n$ is an non-negative integer. If it is not computable, explain why? Question 1.6d of Data structures and algorithm analysis by mark allen wiess. I am able to calculate this sum for n=0,1,2,3.. but I ...
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### Determine if $\sum_{n=1}^{\infty} \frac{5^{n+1}}{(3n-2)!}$ is convergent.

Determine if the following series is convergent: $$\sum_{n=1}^{\infty} \frac{5^{n+1}}{(3n-2)!}$$ so we have $$u_n=\frac{5^{n+1}}{(3n-2)!}$$ $$u_{n+1}=\frac{5^{n+2}}{(3n+1)!}.$$ With the ratio test: ...
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### Taylor expansion of $f(x) = \frac{x}{x+3}\frac{1}{x-2}$ near $x=2$.

I am trying to Taylor expand the function $$f(x) = \frac{x}{x+3}\frac{1}{x-2}$$ around the point $x_0 = 2$. Clearly, the last factor explodes around this point, so I will try and expand that term. ...
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### which denominator will keep the ratio?

I'm given three numbers :$a_1>a_2>a_3$ now I'm told theyre divided by natural numbers $n_1,n_2,n_3$ such that $1>a_1/n_1>a_2/n_2>a_3/n_3>0$ . (the $n$'s could.be any natural numbers ...
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### Geometric progression and logarithms

I would like to ask you for some help, solving that: 'The sum of three members of a geometric progression ($a, aq, aq^2$) is $62$ and the sum of their decimal logarithms $lg$ is equal to $3$. $a$ and ...
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### $\lim_{n \to \infty} \mid a_n + 3(\frac{n-2}{n})^n \mid^{\frac1n} = \frac35$. Then find $\lim_{n \to \infty} a_n$.

Let $\{a_n\}$ be a sequence of real numbers such that $$\lim_{n \to \infty} \mid a_n + 3(\frac{n-2}{n})^n \mid^{\frac1n} = \frac35$$ Then find $\lim_{n \to \infty} a_n$. Tried very hard yet not ...
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### Cantor's theorem about countable sets

Let $f:\mathbb{N} \rightarrow \mathbb{R}$ be a sequence of real numbers. Cantor's theorem states that no interval (a,b) can be the range of a sequence of real numbers. I know the proof for this ...
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### Convergence of $\sum ( \cos \sqrt[3]{n^3 + \sqrt n + 7} - \cos \sqrt[3]{n^3 - 2\sqrt n + 3})$

I have some problem with this example: $$\displaystyle \sum_{n=2}^{\infty}\Bigg(\cos\Big(\sqrt[3]{n^3+\sqrt{n}+7}\Big) -\cos\Big(\sqrt[3]{n^3-2\sqrt{n}+3}\Big)\Bigg)$$ the only idea that crossed my ...
I know that the interval of convergence of the geometric power series $$\sum_{n=0}^\infty x^n=\frac{1}{1-x}$$ is $(-1,1)$. Why is it that if I do the following manipulation $$\frac{1}{1-x}=\frac{1}{... 2answers 80 views ### About bound of series$$ Can anyone please tell me bound for following series I simplified this upto $$(2n-1)-2\left( \frac{1}{n+1}+\frac{2}{n+2}+\ldots+\frac{n-1}{2n-1} \right)$$ Also I get one of my bound as $\frac{n(... 0answers 33 views ### Asymptotic expansion of elliptic integral I am trying to find the first 2-3 terms of the asymptotic expansion in terms of 1/ρ of the elliptic integral I_n(\rho)=\int_0^\frac{h_2}{\rho}\frac{t^{2n}/h_2^{2n}}{(E_n(t))^2\... 0answers 31 views ### Question About Cauchy Product; Is there a formula to multiply many series (More than two) using the Cauchy product? If there isn't, please tell me how I can write this formula$ \left( \frac{1}{a-e^x} \right) ^{n+1}$as the ... 2answers 60 views ### Determine convergence or divergence of the series I have the following series : $$\sum_{k=1}^{\infty }\left ( \frac{k}{\sqrt{4k^{3}+1}} \right )$$ And I am trying to see if it's convergent or divergent. I first thought about the integral test, but ... 0answers 41 views ### Show that$\sum (-1)^n x^{(2^n)}$has no limit as$x \uparrow 1$Show that the following limit does not exist: $$\sum_{0}^{\infty} (-1)^n x^{(2^n)}\text{ with }x \uparrow 1$$ I tried setting $$f(x) = x - x^2 + x^4 - x^8...$$ then $$f(x) = x-f(x^2)$$ then the ... 2answers 43 views ### Prove$f_n(x)=\frac{x^n}{\sqrt{3n}}$for$x \in [0,1]$is uniformly convergent [duplicate] I'm completely confused by uniform convergence, but I put together the following proof just based on my other questions here and examples I read online. Discussion: Let$\epsilon \gt 0$We want to ... 0answers 18 views ### An interesting connection between the Möbius function and the parity of the number of sublattices of index$n$in generic$3$-dimensional lattice I recently discovered an interesting connection between the following two On-Line Encyclopedia of Integer Sequences (OEIS) sequences: A001001 and A209635. More specifically, there seems to be an ... 0answers 64 views ### A series involving digamma function I am trying to solve the series $$\sum_{k=1}^\infty\frac{1}{k(k^2+n^2)}$$ The best I got is $$\frac{\Re\left\{\psi(1+in) \right\}+\gamma)}{n^2}$$ I am not able to simplify it more. Maybe there ... 17answers 14k views ### How can I evaluate$\sum_{n=0}^\infty (n+1)x^n\$
How can I evaluate $$\sum_{n=1}^\infty \frac{2n}{3^{n+1}}$$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...