For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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0
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3answers
33 views

Proof about infinite limsups and subsequences

I am working on a final exam study guide, and came across this question: Suppose limsup$(a_n)$ = $\infty$. Prove: There must exist a sub-sequence ${a_n}_k$ such that ${a_n}_k \to \infty$. My initial ...
2
votes
2answers
39 views

Question about Leonard Gillman's proof of the divergence of the Harmonic series.

Leonard Gillman (1917 – 2009) was an American mathematician, emeritus professor at the University of Texas, Austin. His proof of the divergence of the Harmonic series appeared in The College ...
0
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2answers
23 views

General formula for a summation

I can't find the general formula for the following sum. $q \in \Bbb R, q \ne 1$ $\sum _{i=0}^{n} q^{2i}$ Any hints?
1
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2answers
111 views

Evaluating infinite series $\sum_{n=0}^{\infty} \frac{1}{a^{2}+(2n+1)^2}$

I have no idea to approach this problem. Mathematica gave the sum to be $$ \sum_{n=0}^{\infty} \frac{1}{a^{2}+(2n+1)^2} = \frac{\pi}{4a} \tanh(\frac{a \pi}{2}) $$ How can I analyze this?
0
votes
3answers
76 views

Find a formula for an infinite series

I want to find an explicit formula for $\sum_{n=0}^\infty n^3x^n$ for $|x|\le1$.Is the idea that first to show that this series is convergent and then we can find the number that it converges to? I ...
2
votes
2answers
49 views

Show : $\displaystyle \sum_{n=1}^\infty \dfrac{a_n}{1+a_n}$ is absolutely convergent [duplicate]

Given $\displaystyle \sum_{n=1}^\infty a_n$ is absolutely convergent. Show that $\displaystyle \sum_{n=1}^\infty \dfrac{a_n}{1+a_n}$ also converges absolutely. (If $a_n \neq -1, \forall n \geq 1$ ) I ...
2
votes
4answers
83 views

How do I show that $\sum_{i = 1}^n \frac 1{\sqrt{a_n}} \lt \frac {\sqrt 3}6$ for $a_n = 4n(4n + 1)(4n + 2)$?

Let $a_n = 4n(4n + 1)(4n + 2)$, show that $$\sum_{i = 1}^n \frac 1{\sqrt{a_i}} \lt \frac {\sqrt 3}6 \quad \forall n \in \mathbb{N}^+.$$ I know I need to find an upper bound for $1/\sqrt{a_n}$ but I ...
1
vote
1answer
68 views

Powers-of-10-multiples of $\pi$ (or any irrational) are dense

Very related, but not the same, to this question Multiples of an irrational number forming a dense subset, is the next one: Is the sequence $(\{10^n\pi\})_{n=1}^\infty$ dense in the interval ...
3
votes
2answers
58 views

Evaluate $\lim_{n\to \infty}{\sqrt{(1-\cos(1/n))\sqrt{(1-\cos(1/n))\dots}}}$

Evaluate the limit:$$\lim_{n\to \infty}{\sqrt{(1-\cos(1/n))\sqrt{(1-\cos(1/n))\dots}}}$$ My attempt: let $l$ be equal to that limit so i can write $$l=\lim_{n\to ...
4
votes
3answers
80 views

Why $\displaystyle\sum_{n=1}^\infty {\ln(n^2) \over (n^2) }$ converges?

I have to prove that the series $\displaystyle\sum_{n=1}^\infty {\ln(n^2) \over n^2 }$ converges. The ratio test is inconclusive, so I should use the comparison test, but which series should I compare ...
0
votes
0answers
19 views

On a closed-form from particular values of the Riemann zeta function and divisor functions

I am looking if I can get a closed-form for an infinite series, but I don't know for what it is possible, without finish my computations (see my Question, below). From Applications (8.1 Infinite ...
1
vote
1answer
55 views

Prove that $f(x)=\sum_{n=1}^{\infty} (\frac{x}{n}-log(1+\frac{x}{n}))$ is continuous and can be differentiated ad infinitum

We have $f:(0,\infty) \rightarrow \mathbb{R}$ defined by infinite series $f(x)=\sum_{n=1}^{\infty} (\frac{x}{n}-log(1+\frac{x}{n}))$ Prove that $f$ is continuous and can be differentiated ...
1
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1answer
27 views

Show that $\frac{z}{1-z} = \sum_{j=0}^∞ \frac{2^j}{1 + z^{-2^j}}$ when $z ∈ \mathbb{D}$

The question Knowing that with $z ∈ \mathbb{D}$: $$ \prod_{k=0}^∞(1 + z^{2^k}) = \frac{1}{1-z} $$ prove that with $z ∈ \mathbb{D}$: $$ \sum_{j = 0}^∞ \frac{2^j}{1 + z^{-2^j}} = \frac{z}{1-z} $$ ...
1
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1answer
38 views

Computation of an inverse trigonometric series using complex numbers

The following is a popular question (in competitive exams) in India: Compute the value of $S=\displaystyle \sum_{k=1}^{\infty} \tan^{-1}\left( \dfrac1{2k^2}\right)$. I can compute the value by ...
3
votes
2answers
166 views

Is it true that $log(i) = \frac\pi2i$ ? If so, are both of these legitimate proofs? They seem too beautiful not to be…

Sorry if this is a naive question. I have not yet taken any upper level math courses involving complex numbers. However, in preparation for those courses, together with utilizing the knowledge that ...
0
votes
1answer
73 views

$\sum_{n=1}^{\infty} ne^{-2n}$ estimate to 4 decimal places

I am supposed to estimate the sum correct to 4 decimal places and assume it converges. I know that I am supposed to plug in numbers for $n$ (Instructor says that solving for $n$ is impossible) however ...
0
votes
0answers
11 views

How to interpret $\Phi(z) \sim c_{\kappa}z^{\kappa}+\mathcal{O}(z^{\kappa-1})$

In my reading, I've come across the following statement: The function $\Phi(z)$ is said to have degree $\kappa$ at infinity if $\Phi(z) \sim c_{\kappa}z^{\kappa}+\mathcal{O}(z^{\kappa-1})$ as ...
0
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3answers
27 views

evaluate the following limit else prove that limit does not exist

The function/sequence of interest is as follows: $(\frac{n!}{n!+2})^{n!}$ I have a feeling the limit does exist, as if we divide the numerator and denominator by $n!$ we get ...
1
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1answer
40 views

Finding the general term

I'm having some trouble with trying to find the general term of this sequence. It has a non-linear recurrence. I would really appreciate it if anyone could help me! $ a_{n}= ...
1
vote
2answers
24 views

Let $A\subset \mathbb{R}$ such that $l=\text{inf }(A)$ exists. Prove that $\forall \epsilon >0 $ there is $a\in A$ in the interval $[l,l+\epsilon)$

I need to prove the following: Let $A\subset \mathbb{R}$ such that $l=\text{inf}(A)$ exists. Prove that $\forall \epsilon >0 $ there is $a\in A$ in the interval $[l,l+\epsilon)$ That's what I ...
0
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3answers
70 views

Proving $\frac{n^2}{n-3}$ diverges

I need to prove that $$\frac{n^2}{n-3}$$ diverges For that, I need to prove that, given $\epsilon>0$, we have $n_0$ which depends on $\epsilon$, such that: $$n>n_0 \implies ...
1
vote
3answers
52 views

Uniform convergence of $f_n(x)=\frac{1}{1+nx}$ on $(0,1)$

Consider the sequence of functions $f_n(x)=\frac{1}{1+nx}$ for $x\in (0,1)$. Then $f_n (x) → 0$ pointwise but not uniformly on $(0,1)$. $f_n (x) → 0$ uniformly on $(0,1)$. $\int_{0}^{1} f_n (x) dx ...
2
votes
0answers
25 views

Construct a bounded infinite sequence $x_0, x_1, x_2,$ . . . such that $|x_i − x_j ||i − j|^a ≥ 1$

Given any real number $a > 1$, construct a bounded infinite sequence $x_0, x_1, x_2,$... such that $$|x_i − x_j||i − j|^{a} ≥ 1$$ for every pair of distinct nonnegative integers $i, j$. This means ...
0
votes
1answer
76 views

Show that the series defines and entire function, $\sum_{n=1}^\infty {{z(z+1)\cdots (z+n-1)}\over n^n}.$

Show that the series defines and entire function, $$\sum_{n=1}^\infty {{z(z+1)\cdots (z+n-1)}\over n^n}.$$ I that that an entire function is one that is analytic at every point in the plane. This ...
0
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1answer
35 views

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 ...
-2
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0answers
22 views

Regarding Infinite Sums and their Limit

Suppose a defined Nth partial sum as N tends to infinity: \begin{equation} A_{N}=\sum_{N=0}^{N->\infty} F(N)\end{equation} and its limit $$\lim\limits_{N \to \infty} A_{N}=\lim\limits_{N \to ...
0
votes
1answer
45 views

If $\ (1+3x+x^2)^{10}=\sum_{r=0}^{20}a_r x^r\ $ then…

If $$\ (1+3x+x^2)^{10}=\sum_{r=0}^{20}a_r x^r\ $$ Then then what is the least number except 1 which divides the following:$$\ \sum_{r=0}^{20}(3r+1)a_r\ $$ EDIT: i have put x=1 then it is something ...
2
votes
3answers
64 views

$\sum_{k=0}^{2016}(1+ \omega^k)^{2017}\ $

Let $\omega \ $be a root of the polynomial $\ x^{2016} +x^{2015}+x^{2014}+...+x+1=0 \ $. Then find the value of the following sum: $$\sum_{k=0}^{2016}(1+ \omega^k)^{2017}\ $$ Well I have simplified ...
2
votes
2answers
43 views

Summation of an expression $\sum_{h=0}^{\ln n}\frac{h}{2^h}$

How can be we get the closed form for this expression? $$ \sum_{h=0}^{\ln n}\frac{h}{2^h} $$
-1
votes
0answers
25 views

Sum of a modified Riemann series [closed]

It is well-known that the sum of the classical Riemann series $\sum_{n=1}^{\infty} \frac{\sin((2n-1)x)}{2n-1}$ is $\frac{\pi}{4}(-1)^m$ for every $x$ such that $m\pi<x<(m+1)\pi$, with ...
0
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0answers
22 views

On the zeroes of sine complex function, and a search for a special sequence, following Riemann's approach

If there are no mistakes from the Fourier expansion series for the fractional part function we can write, using a substituion, that for $1<x<e^2$ with uniform convergence $$\frac{1}{2}\log ...
5
votes
1answer
70 views

The number of zeros in the expansion of $n!$ in base $12$

During an interview last year I was asked the following question: How many zeros appear at the end of $n!$ in base $12$, where $n$ is a positive integer? I applied the known Legendre formula for ...
0
votes
1answer
18 views

Can we equate general term of two equal summations provided limit goes from $0$ to $\infty$?

For example: If $∑_{x=0}^{\infty}f(x)=∑_{x=0}^{\infty}g(x)∑_{x=0}^{\infty}f(x)=∑_{x=0}^{\infty}g(x)$, can we say $f(x)=g(x)f(x)=g(x)$ for all $x$, or is there a possibility that they are not ...
0
votes
1answer
40 views

Proof a series is coverges to a specific sum

I need to prove that the sum of the following series is convergent to : $1 \ge Sum$ $$\sum_{n=1}^\infty \ ...
0
votes
1answer
34 views

About intersection of compact sets in non- metric spaces [closed]

Let $X$ be a non- metric space and $\{K_n\}$ be a sequence compact non-empty sets of $X$, with $K_{n+1}\subseteq K_{n}$. Questions: What conditions on $X$ or $\{K_n\}$ imply $\bigcap K_n=\{x\}$ i.e ...
1
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1answer
57 views

Show that $\sum_{k=1}^{\infty}\frac{F_{2^{k-1}}}{L_{2^k}-2}=\frac{15-\sqrt{5}}{10}$

Show that$$\sum_{k=1}^{\infty}\frac{F_{2^{k-1}}}{L_{2^k}-2}=\frac{15-\sqrt{5}}{10},$$ where $F_n$ is a Fibonacci number and $L_n$ is a Lucas number.$^1$ Motivation: For example, when ...
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votes
0answers
99 views

Why is $\sum_{k=1}^\infty -n k^{n-1}=\sum_{k=1}^\infty \left(k^n-(k-1)^n\right)$? [closed]

Why $$\sum_{k=1}^\infty -n k^{n-1}=\sum_{k=1}^\infty \left(k^n-(k-1)^n\right)$$ as well as $$\sum_{k=0}^\infty -n (k+1)^{n-1}=\sum_{k=0}^\infty \left((k+1)^n-k^n\right)$$ always holds in terms of ...
0
votes
1answer
52 views

Any good books for Infinite Series?

I wanted to know about some good or even free books on Infinite Series. Being Poor plz tell me books tht are rather cheap but good books.
1
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3answers
39 views

Is there a formula for finding the nth number in a sequence with a changing pattern

If a sequence has a pattern where +2 is the pattern at the start, but 1 is added each time, like the sequence below, is there a formula to find the 125th number in this sequence? It would also need to ...
1
vote
0answers
35 views

Can we evaluate the alternating sum of the digits of an irrational number?

Suppose you had a summation $\sum(-1)^na_n$, where $a_n$ is the $n$th digit of $e$ and $a_0=2$. I know it diverges, but I want to know if its possible to evaluate anyways. Since it is alternating, ...
1
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0answers
33 views

Tail of a divergent series

We know that if a series converges, its tail tends to $0$ as $n$ goes to infinity. But in the case of a divergent series, what can we conclude about its tail's limit?
3
votes
1answer
39 views

Solving the recursion $p_n = p \cdot p_{n-2} + (1-p)p_{n-1}$

Solving the recursion $p_n = p \cdot p_{n-2} + (1-p)p_{n-1}$ $p_n = p \cdot p_{n-2} + (1-p)p_{n-1}$ $p_n = p \cdot p_{n-2} +p_{n-1} - p\cdot p_{n-1}$ $p_n - p_{n-1} = (-p)(p_{n-1} - p_{n-2})$ $= ...
1
vote
4answers
84 views

Find the limit of $\frac{n^4}{\binom{4n}{4}}$ as $n \rightarrow \infty$

$\frac{n^4}{\binom{4n}{4}}$ $= \frac{n^4 4! (4n-4)!}{(4n)!}$ $= \frac{24n^4}{(4n-1)(4n-2)(4n-3)}$ $\rightarrow \infty$ as $n \rightarrow \infty$ However, the answer key says that ...
1
vote
2answers
30 views

Limit question on independent random variables (Exercise 4.2.4 from Grimmett and Stirzaker)

Let $\{X_r | r \geq 1\}$ be independent and identically distributed with distribution function $F$ satisfying $F(y) < 1$ for all $y$, and let $Y(y) = \min \{k | X_k > y\}$. Show that $$ ...
1
vote
0answers
34 views

Find the function $f(x)$ by using its fourier coefficient

It is easy to find the fourier coefficient and fourier expansion of $f(x)$ function. But I want solve the inverse problem How to find the function $f(x)$, if I know its fourier coefficient (or ...
1
vote
0answers
49 views

A question on sequences using squeeze theorem.

$a_n=(1+\frac{1}{n})^n$ , $b_n=\sum_{k=0}^n \frac{1}{k!}$. Given that $\lim_{n\to \infty}b_n=e$. (i) Show that $b_n-\frac{3}{2n} < a_n < b_n$ for all positive integer $n$. (ii) Using ...
0
votes
2answers
26 views

Showing uniform convergence of a sequence

Let $f_n(x)= \frac x{x+n}$ for $n \in \Bbb{N}$. Show that if $a>0$ then $f_n$ converges to 0 uniformly on $[0,a]$ and show that the convergence is not uniform on $[0,\infty]$. So I've deduced that ...
1
vote
1answer
24 views

Convergence of a series implies convergence of another series

Let $a_1,a_2,\cdots$ be a sequence of real numbers with $a_i\geq 0$. If $\sum_{n=1}^{\infty}\frac{1}{1+a_n}<\infty$ then show that $\sum_{n=1}^{\infty}\frac{1}{1+x_na_n}<\infty$ for each real ...
2
votes
1answer
45 views

Show the series $a_n/(1+a_n)$ converges absolutely

Given that the series $(a_n)$ converges absolutely. Show that the series $(\frac{a_n}{1 + a_n})$ converges absolutely. I am not really sure where to start. Any help would be great.
2
votes
2answers
63 views

Convergence of $\sum _{n=3} ^\infty \frac 1 {(\ln \ln n)^{\ln \ln n}}$

$$\sum _{n=3} ^\infty \frac 1 {(\ln \ln n)^{\ln \ln n}} .$$ I believe the series diverges. I am thinking of using the integral test to show this, but I am not sure if that is right.