For questions about recurrence relations, convergence tests, and identifying sequences

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0
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1answer
62 views

The series $\sum a_nx^n$ with $a_n \ge 0$ for all $n$ and convergent at $R$ also converges at $-R$

Show that if the series converges at $R$,then it also converges at $-R$. What I have done is, since the given power series converges at $R$ (finite quantity), then by $n$-th term test $\lim a_nR^n=0$. ...
1
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0answers
27 views

Series involving Laguerre polynomials

Given the series \begin{align} S_{x}(a) = \sum_{k=1}^{\infty} (-1)^{k+1} \, \binom{x-1}{k} \, L_{k+n-1}(a) \end{align} where $L_{m}(x)$ is the Laguerre polynomial. By using \begin{align} L_{n}(z) = ...
4
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3answers
7k views

Prove: Convergent sequences are bounded

I don't understand this one part in the proof for convergent sequences are bounded. Proof: Let $s_n$ be a convergent sequence, and let $\lim s_n = s$. Then taking $\epsilon = 1$ we have: $n > N ...
0
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1answer
39 views

Why substitution can be used to get Taylor Series?

See the example in: How to find the Maclaurin series for $e^{-x^2}$ If I use $t$ to substitute some function $F(x)$, then the series got still is $t$. So how can I decide under what circumstance the ...
2
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2answers
65 views

Can one differentiate a series after taking its limit?

I made a post in my blog to show that $1+2+3+\cdots=-\frac{1}{12}$. However a visitor of my blog commented that I can not differentiate an expression after taking its limit. Is that statement true? ...
1
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2answers
81 views

How to derive the formula for the sum of the first $n$ perfect squares? [duplicate]

How do you derive the formula for the sum of the series when $S_n = \sum_{j=1}^n j^2$? The relationship $(n, S_n)$ can be determined by a polynomial in $n$. You are supposed to use finite differences ...
3
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0answers
42 views

Is this function, a sum of one term and a convergent series, analytic?

$$(\frac{1}{z} + \sum z^n)$$ for 0<|z|<1. This is for complex variables. So, the series, convergent for the above domain of definition, always represents an analytic function. What about the ...
0
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1answer
56 views

Infinite sum with an improper integral

Trying to find the variance of the logistic distribution I have come to this expression. I tried to solve the integral separately, but this procedure is not working. Can someone guide me on how to ...
0
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2answers
57 views

limit inside of an infinite sum

How to solve this expression? \begin{equation} 2\sum^{\infty}_{n=1}-\frac{n^{2}(-1)^{n-1}}{n^{3}}\lim_{p\rightarrow \infty}\frac{np^{2}+n2p+2}{e^{np}}=? \end{equation} This expression tends to zero ...
-1
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0answers
60 views

sum of the series and infinity, again

If you have the sums $f(n) = 1^{59} + 2^{59} + 3^{59} + \cdots + (10^n)^{59}$ and $g(n) = 1^{5} + 2^{5} + 3^{5} + \cdots + (10^n)^{5}$, for large enough $n$, $f(n)$ is approximately $\frac{1}{60} ...
1
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1answer
70 views

Determine convergence of the series $\sum\limits_{n=2}^{\infty}\frac{\cos(\ln(\ln(n))}{\ln(n)}$

How to determine convergence of the series $\sum\limits_{n=2}^{\infty}\frac{\cos(\ln(\ln(n))}{\ln(n)}$ ? I tried to get somewhere with Integral criteria and with comparing to other series but ...
-1
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1answer
37 views

Explain proof of sum equivalence

I need to prove: $$\sum_{k=1}^n k{n \choose k} 2^{k-1} = n3^{n-1}$$ I have the answer, but I can't understand how can I get from step 1 to step 2?! Step 1: $$(1+x)^n = \sum_{k=0}^n {n \choose k} ...
4
votes
1answer
87 views

How is this series evaluated?

I entered this series into Mathematica to see if it could be simplified and it managed to give a form in terms of the partial gamma function. However I do not know how it derives this formula and I am ...
0
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1answer
57 views

Why $\sum\limits_{n=1}^\infty \frac{e^{inx}}{n}=-ln(1-e^{ix})$ in $D'$

Functions in D are finite test functions in $C^\infty(\mathbb{R})$ D' are distributions (genralized functions) Do I have to check that $\forall \phi \in D$: $\lim_{\epsilon \to 0} ...
2
votes
3answers
43 views

Prove that for any positive integer $n$ and $d$, $\sum_{k=0}^d 2^k\log_2(\frac{n}{2^k})=2^{d+1}\log_2(\frac{n}{2^{d-1}})-2-\log_2{n}$

I could prove it by induction, but I need to see how I might have discovered it for myself (cause that's what's gonna be on exam).
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4answers
38 views

How to explain that function has positive and negative values around zero?

I have following function $$f(x)=\begin{cases} x^2\cos\left(\frac1x\right) &\text{if }x\neq0\\ 0 &\text{if }x=0 \end{cases}$$ How can I prove that this function in every area of zero has ...
5
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1answer
39 views

Can anyone prove D'Alembert Criterion (Dalambert) criterion for converging positive sequences?

This will most likely be on the exam, but it is not given in the text book. In my notebook I have this proof which I will type out, but it makes no sense. Here it goes: $$\text{D'Alembert ...
0
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3answers
50 views

Prove that the elements of these two sequences are not null

Let $x_{n+1}=x_n+2y_n$ and $y_{n+1}=y_n-x_n$, where $x_1=1$ and $y_1=-1$. I tried proving by contradiction, I tried by induction, I got nothing. This is a question I had on an exam, I didn't manage ...
3
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2answers
43 views

When are both $\sum_{n=0}^\infty \log(a_n)$ and $\sum_{n=0}^\infty a_n$ convergent?

I'm new to this site. Can someone give me some examples of when both: $$\sum_{n=0}^\infty \log(a_n)\qquad \text{ and }\qquad \sum_{n=0}^\infty a_n$$ are convergent?
1
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3answers
62 views

Showing that $a_n = 1-(\frac{-2}{3})^n$ is not bounded

I have the sequence $a_n = 1-(\frac{-2}{3})^n$, and I need to show if it is bounded. I was first under the assumption that, if a sequence is monotonically increasing/decreasing and bounded that the ...
1
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5answers
52 views

Show that this limit is related to Euler number

I am calculating the limit $\lim_{n \rightarrow \infty} \left( \frac{n!^{\frac{1}{n}}}{n} \right)= \frac{1}{e}.$ I got this limit from wolframalpha, but don't know how to show this.wolframalpha
0
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1answer
26 views

Analysis Doubt on sequence and series of functions

I have seen in Rudin the following "if a compact class of bounded continuous functions on a compact metric space is not equi-continuous then that class contains a sequence which has no equi-continuous ...
7
votes
3answers
311 views

Solve infinite series equation with logarithmic terms.

Solve logarithmic equation: $$\frac{\log x^2}{\log^{2}x}+\frac{\log x^3}{\log^{3}x}+\cdots+\frac{\log x^k}{\log^{k}x}+\cdots=8$$ here $\log$ is assumed to have base $10$. So far I managed to rewrite ...
2
votes
3answers
91 views

Find all $x$ such that $2^x,2^{x^2}$ and $2^{x^3}$ form $3$ terms of an A.P.

I know that if $a,b,c$ are in Arithmetic Progression, then $2b=a+c$, but in this case, I am unable to solve for $x$. Hints are appreciated. Thanks.
1
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0answers
22 views

Converse of uniform convergence theorem

Let $f(x) = \sum_{i=1}^\infty f_i(x)$. Suppose that $f$ is continuous, and each $f_i$ is continuous. Does it follow that the series converges uniformly to $f$?
14
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3answers
290 views

How to compute $\sum_{n\text{ odd}}\frac{1}{n\sinh n\pi\sqrt 3}$?

I came across an old question asking to show that $$\displaystyle\sum_{n\text{ odd}}\frac{1}{n\sinh n\pi}=\frac{\ln 2}{8}.\tag{1}$$ Although I have managed to prove this formula, my proof uses ...
0
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1answer
48 views

proof that $\limsup a_n=\sup\{a_n,a_{n+1},…\}$

How can I prove that $\limsup a_n= \sup\{a_n,a_{n+1},...\}$? I also need to prove: for two sequences $a_n>0$ and $b_n \ge 0$, then $\limsup(a_n b_n) \le \limsup(a_n) \limsup(b_n)$. I thought ...
0
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0answers
45 views

Factorial Series I

Given the factorial series \begin{align} \beta_{1}(x) = \sum_{s=1}^{\infty} \frac{(-1)^{s} \, s!}{x(x+1)(x+2)\cdots(x+s)} \end{align} then what are the coefficients, $a_{s}$, of the square of ...
2
votes
1answer
295 views

Uniform convergence of $f_n = (n^a x^2)/(n^2 +x^3)$

My question is, if you have the sequence $$f_n = \frac{n^\alpha x^2}{n^2 +x^3}$$ on $[0, \infty)$, for values of a for $0<\alpha<2$ does the sequence uniformly converge? I guess another way to ...
0
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1answer
43 views

Bound of Mann iterative sequence

There is theorem in the book of Charles Chidume "Geometric Properties of Banach Spaces and Nonlinear Iterations" My question is: why if the underlined conditions are satisfied {Xn} is bounded (proof ...
0
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1answer
25 views

Limit of summation as n goes to infinity

I am trying to solve the following: Let $q>1$ and $n \in N$. Evaluate $\lim_{n \rightarrow +\infty} \sum_{k=1} ^n \frac{k^{q-1}}{n^q + k^q}$. I understand that I need to first get the summation ...
6
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1answer
164 views

How do I evaluate this sum :$\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$?

How do I evaluate this sum :$$\displaystyle\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$$ Note : I used many criterions of convergence to show if it converges but i didn't up. Thank you for any ...
9
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2answers
1k views

find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges

Find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges. To make it more challenging, produce examples where $a_n$ and $b_n$ are ...
0
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1answer
48 views

Sum of a series converges iff the sum of a function of the series converges

I am trying to understand the concept that a sum of a positive series converges iff the sum of a function of the series converges, i.e. $\sum a_n$ converges $\iff$ $\sum f(a_n)$ converges for $a_n ...
0
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0answers
42 views

How to determine Taylor series expansion of function $f(x) = \frac{\cos(x)}{x}$ about $a=1$?

Given function is $f(x) = \frac{\cos(x)}{x}.$ $y = x - a , y = x - 1$. $x = y+1 , f(y) = \frac{\cos(y+1)}{y+1}$ How to get Taylor series expansion about $1$ of this function? If it was needed to ...
2
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0answers
19 views

“Complement” of Kempner Series

It is a long time since I summed any series. I was aware that the harmonic series diverged, (if I recall you can keep making groups that are greater than a half). Then today I saw SMBC and it blew my ...
0
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1answer
16 views

Summing Bases and Comparing

Let $b$ be an integer greater than 2, and let $N_b = 1_b + 2_b + \cdots + 100_b$ (the sum contains all valid base $b$ numbers up to $100_b$). Compute the number of values of $b$ for which the sum of ...
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1answer
24 views

$u=\sum (u,\varphi_k)_{L^2}\varphi_k$ vs $u(x)=\sum (u,\varphi_k)_{L^2}\varphi_k(x)$ and related questions

I have some doubts after reading several threads. Let us work on a bounded domain $\Omega$ with Neumann BCs, with $\varphi_k$ and $\lambda_k$ being the orthonormalised eigenvectors and eigenvalues of ...
29
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6answers
2k views

How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists?

I would like to investigate the convergence of $$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt\ldots}}}}$$ Or more precisely, let $$\begin{align} a_1 & = \sqrt 1\\ a_2 & = \sqrt{1+\sqrt2}\\ a_3 ...
1
vote
1answer
27 views

Convergence of subsequence of partial sums implies full convergence?

Define $S_n = \sum_{j=1}^n a_j$ a sum of real numers. Suppose there is a subsequence such that $S_{n_j} \to S$ for some $S$, i.e., the infinite sum converges along this subsequence. Does this imply ...
3
votes
4answers
698 views

Alternate method to calculate an infinite string of numbers that's not $\pi$, and contains any string

So, rather than using $\pi$, is there any way that isn't overly complicated, (and can be calculated on a computer without taking a year) in which I could generate an infinite string of numbers that ...
1
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3answers
94 views

Taylor series expansion of function $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}}$ [duplicate]

Determine the Taylor series expansion of function $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}}$. $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}} = \arcsin(x)\frac{1}{\sqrt{1-x^2}}$ It is known: (1.) ...
2
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2answers
146 views

Does this weird series converge?

$\sum_{n\in S}$$\frac{1}{n}$, where S consists of those positive integers whose decimal expansion does not contain the digit 1. This was a part(b) question. Part (a) was an evaluation of the ...
0
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1answer
21 views

How can I count the number of $n$ digit positive integers without a specific digit?

Came across the Kempner Series and was doing a little reading. The proof that the Kempner Series is bounded by 80 requires the fact that the number of $n$ digit positive integers without the digit 9 ...
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votes
2answers
56 views

$\sum a_n$ converges $\iff$ $\sum f(a_n)$ converges [duplicate]

For specific functions and $a_n >0$, we can say that $\sum a_n$ converges $\iff$ $\sum f(a_n)$ converges. I want to show this specifically for the case that $f(x)=sin(x)$. This is what I have thus ...
1
vote
3answers
61 views

Find the sum of first $20$ terms of a sequence

Define a sequence $$a_n=\sqrt{1+\left(1-\frac{1}{n}\right)^2}+\sqrt{1+\left(1+\frac{1}{n}\right)^2}$$ for $n \geq 1$. Find $$\sum_{i=1}^{20}\frac{1}{a_i}$$ Some insight on the approach is highly ...
1
vote
2answers
162 views

How was the equation re-written?

This question is a part of inhomogeneous recurrence relations (IHR). The actual question was Find a solution to $a_n - a_{n-1} = 3(n-1)$ where $n \ge 1$ and $a_0 = 2$. While going through the ...
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0answers
16 views

What is the series for water retention patterns in a Pascal type triangle [closed]

The number of different water patterns for a square with open boundaries is A054247 Number of n X n binary matrices under action of dihedral group of the square D_4. +10 9 1, 2, 6, 102, ...
1
vote
0answers
23 views

Solve one dimensional wave equation using fourier transform

I'm trying solve this wave equation using fourier method, but I am stuck... $${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \ 0<x\le L, t >0 $$ $${ u }( 0,t) = { u }( L,t) = 0$$ $${ ...
-3
votes
1answer
44 views

Find the nth term of geometric series

Given the geometric series $1+2+4+8...$ Find the sum between (inclusively) the 5th term and the 15th term. I just solved for the 5th term. $r=2$ so just multiply the 4th term $8\cdot2$ to get 16. ...