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

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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 ...
0
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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 ...
1
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1answer
27 views

Probability of Limsup of a bunch of events

In "Probability Theory" by Athreya, I encountered the following question: Let $\{A_n\}_{n=1}^\infty$ be a sequence of events on a common probability space. Suppose $$\sum_{n=1}^\infty P(A_n\setminus ...
5
votes
4answers
225 views

Is it possible to work out the derivative of $e^x$ using the summation definition of $e = \sum_n 1/n!$?

So I know this question is a bit obtuse because usually we define $e$ in terms of the $\lim_{n \to \infty} (1 + 1/n)^n$ definition, and then compute derivatives of $e^x$ from there appealing to the ...
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?
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 ...
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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 ...
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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
1
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0answers
95 views
+50

Integrating an infinite sum (statement from paper)

Let us work on a bounded domain $\Omega$ with Neumann BCs, with $\varphi_k$ and $\lambda_k$ being the orthonormalised eigenvectors and eigenvalues of the Neumann Laplacian. Define $$v(x,y) = \sum_{k ...
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 ...
5
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3answers
141 views

Prove that $A_{100} \gt 14$ where $A_{n}=A_{n-1}+\frac{1}{A_{n-1}}$ and $A_1=1$

I tried attempting the question, and the best upper bound I could obtain was $1+\ln{98}$. I tried using $A_{n}\le n$ to form a harmonic series, but that wasn't strong enough. Any help would be ...
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 $f$?
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 ...
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 ...
7
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3answers
313 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 ...
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 ...
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 ...
1
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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 ...
1
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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 ...
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 ...
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.) ...
1
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3answers
62 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
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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... $${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \ 0<x\le L, t >0 $$ $${ u }( 0,t) = { u }( L,t) = 0$$ $${ ...
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.
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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. ...
3
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4answers
700 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 ...
-3
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1answer
33 views

find the common ratio of geometric series [closed]

The sum of first $3$ terms of a geometric series is $39$. If $t_1=27$ ,what is the common ratio? look for all possible answers.
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 ...
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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 ...
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 ...
0
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1answer
12 views

Absolute/conditional convergence problem

I am trying to determine whether $\sum_{n=1} ^\infty (-1)^n \frac{3n}{\sqrt(n^3 + 2)}$ is absolutely convergent, conditionally convergent, or if it diverges. Thus far, I've shown that $\sum_{n=1} ...
0
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1answer
46 views

Spivak's Limit Comparison Test proof

On page 476 in Spivak's Calculus he leaves a detail in his Limit Comparison Test proof to the reader and I was just curious as to whether or not I'm showing it correctly. With 2 sequences $a_n$, $b_n ...
0
votes
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 ...
2
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1answer
46 views

Expressing e as an infinite series: finding values for similar series

I am supposed to be using the fact that $e = \sum_{n=0} ^\infty \frac{1}{n!}$ to find the value of $\sum_{n=0} ^\infty \frac{1}{2n!}$. Is there some method for substitution when dealing with infinite ...
1
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1answer
43 views

Greatest integer function in infinite sum of geometric series

I am not sure how to approach this infinite sum. I can see that it's a geometric series and that has a straightforward solution, but I am not sure how to address the alternating signs with the ...
3
votes
1answer
21 views

Question regarding convergence in $L^p$ spaces.

When solving an exercise regarding $\ell^p$ spaces, I came up with the following question. The exercise said, Let $1<p\leq \infty$ and let $p'=\frac{p}{p-1}$. Let $b\in \ell^{p'}$ and define ...
1
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1answer
32 views

For which values of $x \in I$ is $(f_n)$ differentiable term by term?

Let $f_n(x) = \frac{1}{n}e^{-nx}$ on $x\in[0,1] =I$. Discuss the pointwise and uniform convergence of $(f_n)$ on $I$. For which values of $x \in I$ is $(f_n)$ differentiable term by term? ...
3
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2answers
56 views

Calculate the sum $\sum_{n=0}^\infty\frac{1}{(4n)!}$ [duplicate]

How to determine the sum $\sum_{n=0}^\infty\frac{1}{(4n)!}$ ? Do I need to somehow convert (4n)! to (2n)! or in tasks like this, should I get the (4n)! after some multiplying? Thank you all for your ...
2
votes
4answers
233 views

calculate the $1/6+1/12+1/24+1/48 \ldots $. Wolfram is wrong?

I am trying to calculate the following sum $$ S = \frac{1}{6}+\frac{1}{12}+\frac{1}{24}+\frac{1}{48} + \cdots $$ so $$ S+\frac{1}{3} = \frac{1}{3} + \frac{1}{6}+\frac{1}{12}+\frac{1}{24}+\frac{1}{48} ...
5
votes
3answers
75 views

Infinite sum of alternating telescoping series

I am struggling to find the sum of the following series: $$\sum_{n=1} ^{\infty} \frac{(-1)^n}{(n+1)(n+3)(n+5)}.$$ It seems as though it should be a straightforward telescoping series. I attempted to ...
7
votes
1answer
80 views

Does the minimum of two decreasing divergent series diverge? [duplicate]

If $\displaystyle \sum_{n=1}^{\infty}a_n$ and $\displaystyle \sum_{n=1}^{\infty}b_n$ are both divergent series with $a_n\downarrow0$ and $b_n\downarrow0$, [so $(a_n)$ and $(b_n)$ are decreasing ...
1
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0answers
15 views

Uniformly Bounded Sum of Functions

I know that $\sum_{j=1}^{N}\sin(jx)$ is uniformly bounded because I can express $\sin(jx)$ in terms of exponentials and use properties of a geometric series. However, I am having a very difficult ...
2
votes
2answers
59 views

Product of Matrices I

Given the matrix \begin{align} A = \left( \begin{matrix} 1 & 2 \\ 3 & 2 \end{matrix} \right) \end{align} consider the first few powers of $A^{n}$ for which \begin{align} A = \left( ...
1
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1answer
28 views

How do I find the interval of convergence?

Suppose I have: $$\sum \cfrac{(-1)^n}{\sqrt{n}}x^n$$ If I use the ratio test, I get $$\cfrac{1}{\sqrt{1+\frac{1}{n}}}|x|$$ Why can it be said the radius of convergence here is $1$? Disregard the ...
0
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1answer
63 views

Gosper Formula for inv $\pi$, properties.

I need to understand very good how the properties of this formula $\frac{4}{\pi} = \frac{5}{4} + \sum_{N \geq 1} \left[ 2^{-12N + 1} \times(42N + 5)\times {\binom {2N-1} {N}}^3 \right] $ Taken from ...
2
votes
5answers
65 views

Find the infinite sum of a sequence

Define a sequence $a_n$ such that $$a_{n+1}=3a_n+1$$ and $a_1=3$ for $n=1,2,\ldots$. Find the sum $$\sum_{n=1} ^\infty \frac{a_n}{5^n}$$ I am unable to find a general expression for $a_n$. Thanks.
1
vote
1answer
24 views

Infinite matrix is injective if all its upper left minors are invertible?

This is a natural generalization of a recent MSE question. Let $X=(x_k)_{k\geq 1}$ be a sequence of real numbers, and $A=(a_{ij})_{i\geq 1,j\geq 1}$ be a real infinite matrix indexed by ${\mathbb ...
2
votes
4answers
152 views

How to get the given equality?

I have the following sum ($n\in \Bbb N)$: $$ \frac {1}{1 \times 4} + \frac {1}{4 \times 7} + \frac {1}{7 \times 10} +...+ \frac {1}{(3n - 2)(3n + 1)} \tag{1} $$ It can be proved that the sum is equal ...
1
vote
1answer
42 views

$\sum 0$: does it converge or diverge?

Sometimes I have to do exercise with parameter and, if I substitue particular value of the parameter, I obtain $\sum_{n=1}^{\infty} 0$. But it isn't clear for me if in this case the series converges ...