# 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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### For which $s$ is defined $\frac{1}{\zeta(s)}\sum_{n=1}^\infty\frac{n}{ \left( \zeta(s) \right) ^n}$, where $\zeta(s)$ is the Riemann Zeta function?

If there are no mistakes in my words, if one of the factors $\sum_{n=1}^\infty a_n$ of a convolution product or Cauchy product converges, and the other $\sum_{n=1}^\infty b_n$ corverges absolutely ...
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### What is the coefficient and constant term in the following sequence defined recursively?

Let $f_n(x)$ be a sequence of polynomials defined inductively as $f_1(x) = (x - 2)^2$ $f_{n+1}(x) = (f_n(x) - 2)^2$ $; n \ge 1$ Let $a_n$ and $b_n$ respectively denote the constant term and the ...
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### bounded but not convergent sequences

I am not sure that if this question has a positive answer...I am looking for a sequence of real numbers $(p_{n})_{n\geq 1}$ such that $-1<\lim _{n}\inf p_{n}\leq \lim_{n}\sup p_{n} <1$ (as ...
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### Series proof needed

I have following equations but I do not know the proof. Kindly provide the proof or give me some reference to look into. Here are the equations. 1- ...
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### Why is it that when n ≥ 1 the series is $\le$ 1/4 [closed]

So how is the series $\sum_{n=1}^\infty \frac{1^2 * 3^2 * 5^2 ... (2n-1)^2}{2^2 * 4^2 * 6^2 ... (2n)^2}$ < 1/4 for n $\ge$ 1 is it because the series is divergent outside of the interval of ...
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### show function has a uniformly convergent subsequence

alright, so I have this question from my analysis class and I believe I have answered it correctly. I would be grateful if you could read it and give me your thoughts. A sequence $f_n$ of real valued ...
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### How to show that all sequentially compact spaces are bounded?

I want to show given a sequentially compact subset $A \subseteq M \implies A$ is bounded. I read this Every sequentially compact set is closed and bounded. but the proof is poorly written and jumpy ...
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### Does equality of the sum of two such series imply equality of each term of that series?

Let a(1)< a(2) < ..< a(m) and b(1)< b(2)<..< b(n) be real numbers such that $$\sum_{i=1}^m |a(i)-x| = \sum_{j=1}^n |b(j)-x|$$ for all x belonging to R. Show that m=n and ...
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### When adding or subtracting two infinite sums, why is there no issue with “staggering” or arbitrarily manipulating the “alignment” of terms?

I was watching Ramanujan: Making sense of 1+2+3+... = -1/12, where the presenter writes: (I tried to write this out in $\LaTeX$ but couldn't figure out how to do multi-column alignment without ...
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### Proof of the Strenghtened Limit Comparison Test

I'm studying on my own using Bonar and Khoury's Real Infinite Series. I understand the proof of the "regular" Limit Comparison Test( a link to google books, page 23 ) but the book doesn't provide a ...
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### $\sum_{k=1}^n 1/k - \log n$

I got this question : $$a_n = \sum_{k=1}^n \frac 1k - \log n$$ I proved that $\lim a_n$ exist. Now I have to prove: $$0<a_n-\lim a_n\le \frac 1n$$ for every $n \in \mathbb N$. I tried ...
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### Determine wether $\sum_{n=1}^\infty \frac{3n-1}{(n+1)^3}$ converges or diverges.

Determine wether the following function converges or diverges by comparison test: $\sum_{n=1}^\infty \frac{3n-1}{(n+1)^3}$ Upon inspection I can clearly see that the series converges. However I am ...
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### Is there a general approach to find an explicit formula to a recursive sequence?

I've been doing a bunch of exercises where I need to find the explicit formula for a given sequence... $$a_{n+2} = 6 a_{n+1} - 9 a_{n}$$ The first were easy with $a_{0} = 1$ and $a_{1} = 3$ i.e.. But ...
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### How is the last “=” true?

How can the last equality be true? $$G(u)=\frac{g}{(1+u)^g-1}-\frac1u=\frac{g}{gu + \cdots + u^g} - \frac{1}{u}=\sum_{k=0}^\infty c_ku^k$$
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### Finding the nth term of a geometric series [closed]

Please help me find the nth term of the series: $$\frac{2}{5}-\frac{6}{5^2}+\frac{10}{5^3}-\frac{14}{5^4}+\cdots$$
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### $\lim a_n = L \implies \lim a_n^2 = L^2$

I have to prove the following: $$\lim a_n = L \implies \lim a_n^2 = L^2$$ I know that $\lim a_n = L \implies \forall\epsilon>0 \ \exists n_0$ such that $n>n_o\implies |a_n-L|<\epsilon$ I ...
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### series convergence help using tests [closed]

Use any theorems or properties of series. This was a question on my homework and I received 0 points. I need help with the entire question. I originally tried to compare part (a) to the ...
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### How to prove $\sum n/3^n$ converges without ratio test?

The only tests my class has learned so far and is allowed to use are: Divergence, Integral, Comparison, Geometric/Harmonic/Telescopic. I have proved that the series converges, via a ratio test, but my ...
### Solve for bound of $\sigma(n)$ from harmonic series.
I am given the harmonic series, $H_n$. How can I show that for $n\geq 1$, $$\sigma(n)\leq H_n+e^{H_n}\ln{H_n}$$ By the way, if you're not familiar with it, $\sigma(n)$ is the sum of all positive ...