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

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3
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5answers
66 views

A limit problem: $\lim\limits_{n\to\infty}\frac{1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^n} }{1+\frac{1}{3}+\frac{1}{9}+\cdots+\frac{1}{3^n} }$

I need help in solving the limit below: $$\lim_{n\to\infty}\frac{1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^n} }{1+\frac{1}{3}+\frac{1}{9}+\cdots+\frac{1}{3^n} }$$ What I've done is to simplify ...
0
votes
3answers
60 views

Express the number $4$ and $5$ and $6$ and $7$ and $8$

Express the number $4$ and $5$ and $6$ and $7$ and $8$ with trigonometric identities or series or equations. example: Express the number $1$, $$\cos^2 x + \sin^2 x=1$$ Express the number $2$, ...
0
votes
0answers
16 views

Infinite sub-sequences that make up a sequence

A sequence $\{a_n\}$ can be broken into sub-sequences, $\{a_n\}^1_{k_1}, \{a_n\}^2_{k_2}, \dots,\{a_n\}^m_{k_m}$, if every element in $\{a_n\}$ belongs to at least one of the sub-sequences. I had to ...
0
votes
1answer
35 views

How can I raise a Taylor Series to a power?

I have recently been undertaking the challenge of finding the antiderivative of $x^x$. In doing so, I have come across the idea of raising a Taylor series to a variable exponent. I came to the ...
1
vote
1answer
58 views

Definition of exponential function -

A lot of textbooks offer a definition of the exponential function such as this: $$\exp(x):=\sum_{k=0}^{\infty}\frac{x^k}{k!}.$$ a) Show that the given definition for $\exp$ is correct, ...
1
vote
1answer
28 views

Integral approximation for alternating series

I can approximate the sum of $\frac 1 {n^2}$ using its integral. But what about $(-1)^n\frac 1 {n^2}$? Is it possible to approximate this using integrals? I want to know if there are other ways than ...
1
vote
4answers
37 views

Closed set in $l_{2}$

I need to show that the set $A=\left\{ x \in l_{2} : |x_{n}| \leq \frac{1}{n}, n=1, 2, ...\right\}$ is a closed subset of $l_{2}$ I'm assuming the best way to show this is to have a sequence in A ...
3
votes
1answer
33 views

Analytic continuation of a certain Dirichlet series

Is there an elementary way to analytically continue $$f(s)=\sum_{n=1}^\infty \frac{(-1)^n}{(2n+1)^s}$$ to the entire complex plane? It is not hard to see (by grouping terms in pairs and using the ...
10
votes
4answers
240 views

Is there a function that can be subtracted from the sum of reciprocals of primes to make the series convergent

The gamma constant is defined by an equation where the harmonic series is subtracted by the natural logarithm: $$\gamma = \lim_{n \rightarrow \infty }\left(\sum_{k=1}^n \frac{1}{k} - \ln(n)\right)$$ ...
2
votes
3answers
62 views

Calculating $\sum_{k=0}^{n}\sin(k\theta)$ [duplicate]

I'm given the task of calculating the sum $\sum_{i=0}^{n}\sin(i\theta)$. So far, I've tried converting each $\sin(i\theta)$ in the sum into its taylor series form to get: ...
7
votes
1answer
83 views

Does $\tan (x)$ equal $\frac{-1}{x-\frac{\pi}{2}}+\frac{-1}{x+\frac{\pi}{2}}+\frac{-1}{x-\frac{3\pi}{2}}+\frac{-1}{x+\frac{3\pi}{2}}+…$?

I set my Year 12 students a question involving the sums of rational functions $\frac{1}{x-n}$. The graph of a sum of these functions looks an awful lot like a tan graph. This led me to ask: Does ...
3
votes
0answers
49 views

Limit of a sum.

While fixing my answer to this question I noticed that (actually the question is equivalent to this modulo some algebra) $$\frac{1}{2}=\lim_{x\to\infty}\sum_{i=0}^\infty ...
4
votes
3answers
75 views

How do I evaluate $\lim_{n\rightarrow \infty}\frac{1}{\sqrt{n}}\sum_{k=1}^{n}\frac{1}{\sqrt{k}}$?

How do i evaluate this limit : $$\displaystyle \lim_{n\rightarrow \infty}\frac{1}{\sqrt{n}}\left(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+............+\frac{1}{\sqrt{n}}\right)$$ ? Thank you for any ...
0
votes
1answer
44 views

Bizarre definition of convergent sequences

I've recently begun studying Analysis, specifically the sequence and series part. I've just come across the definition of convergence: A sequence $(a_n)$ converges to a real number $a$ if, for ...
0
votes
1answer
53 views

What is the function $f$ such that $\sum_{k=0}^n f(k)=n^3$?

$$\begin{align*} 1 &\leadsto 1 \\ 1+3 &\leadsto 2^2 \\ 1+3+5 &\leadsto 3^2 \end{align*}$$ In general, if $f(x)=2x+1$, then $f(0)+f(1)+f(2)...f(n)=(n+1)^2$. Now, $$\begin{align*} 1 ...
5
votes
3answers
138 views

Closed form for a binomial series

I am wondering if any knows how to compute a closed form for the following two series. $$\sum_{m=1}^{n}\frac{(-1)^m}{m^2}\binom{2n}{n+m}$$ $$\sum_{m=1}^{n}\frac{(-1)^m}{m^4}\binom{2n}{n+m}$$ ...
0
votes
1answer
36 views

Multiplication of 2 sums that equal another multiplication of 2 sums

I have been trying to prove a formula of mine and i come across something very interesting, well to me it is. If the formula is correct, it states that: $$ \left(\sum_{m=0}^{k-c} {k-c \choose m}{ms_1 ...
2
votes
5answers
84 views

Can anyone prove the identity $\sum_{m=-\infty}^\infty (z+\pi m)^{-2} = (\sin z)^ {-2} $

I came across this identity in a paper on elliptic curves, and the proof wasn't provided. It really irked me, and I couldn't find an explanation anywhere else. Can anyone shed some light? ...
-3
votes
2answers
93 views

What is $\sum_{n=0}^\infty n^n$ [on hold]

If it does not work above: 1^1+2^2+3^3+4^4...= to what? If the answer is found, then what would be the proof of it and why? Although it does diverge, some divergence sums amount to a number < ...
0
votes
0answers
14 views

What formula would I use for a four factor prioritization method where the factors are summed and ranked?

We are developing a way to prioritize system issues. Our current ranking is 1 - 5, but that becomes rather flat when dealing with a couple hundred issues. In our new method, we have four factors in ...
1
vote
4answers
87 views

If $x_{n+1}\leq x_n + 1/n^2$ then $x_n$ converges [duplicate]

Let $x_n$ be a sequence of non-negative real numbers such that $\forall n, x_{n+1}\leq x_n+ \frac{1}{n^2}$ Prove that $x_n$ converges. The problem is trivial whenever $x_{n}$ is an ...
2
votes
3answers
73 views

value of an $\sum_3^\infty\frac{3n-4}{(n-2)(n-1)n}$

I ran into this sum $$\sum_{n=3}^{\infty} \frac{3n-4}{n(n-1)(n-2)}$$ I tried to derive it from a standard sequence using integration and derivatives, but couldn't find a proper function to describe ...
0
votes
1answer
68 views

Seemingly simple logic question

I found this pleasant textbook on Proof Theory online and free: Introduction to Proofs, an Inquiry-Based approach To quote (page 9): 2.26 DEFINITION. A sequence $\langle x_0,x_1, . . . ...
0
votes
2answers
36 views

Proving a Recursive Formula

I know there are some questions on this site about how to find a recursive formula, but I've already found the formula. I'm doing an assignment (http://mathstat.dal.ca/~svenjah/math2112/Assign6.pdf) ...
0
votes
2answers
46 views

Give the three numbers that form a geometric sequence. [on hold]

Three numbers form a geometric sequence. If 5 is added to the second term, then the resulting numbers will constitute an arithmetic sequence. If 22.5 is added to the third number, these numbers will ...
3
votes
2answers
205 views

How to prove the non-existence of a polynomial series that uniformly converges to this function?

I was asked to prove the following. There exists a function $p(x)\in C(a,b)$ $(-\infty<a<b<+\infty)$, such that there does not exists a polynomial series which uniformly converges to ...
0
votes
2answers
26 views

Excel's EXP function compared to a series expansion

I am comparing the results of a series expansion of $e^x$ to Excel's $\mathop{EXP}(x)$ function. Should I expect them to be the same? Excel's gives $\mathop{EXP}(10) = 22026.4657948067$. However, ...
0
votes
1answer
36 views

By what rule can't you do this specific action with respect to infinite sums? [on hold]

An example of this is the summation of 1+2+3...=-1/12. By some reason, you cannot change the digits of that to 1+(1+1)+(1+1+1)... which would be equal to -1/2. -1/12 is not equal to -1/2 though.
1
vote
1answer
75 views

Sum of trigonometric infinite series

I am trying to prove that for any $x\geq 1$ we have: $$ \sum_{m=1}^{\infty} \frac{\sin\frac{(2m-1)\pi}{x}}{\left(\frac{(2m-1)\pi}{x}\right)^3} = \frac{x}{8}(x-1). $$ Could I have some help please? I ...
0
votes
3answers
95 views

Sum of $\sum\limits_{x=-\infty}^{\infty}x^{\operatorname{sign}(x)}$

Both the sum of $1+2+3+4+\cdots$ and the sum of $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots$ diverge. If both are paired together in one function, as seen above, can they amount to a ...
3
votes
3answers
70 views

“fast enough” decay of an $\ell ^2$ sequence implies $\ell ^1$?

To be specific, say we are given that $(a_n)$ is a sequence of real numbers such that \begin{equation} \sum_{n=1}^{\infty} n^3 a_n^2 < \infty. \end{equation} Is it then true that $$ ...
1
vote
2answers
63 views

How to prove that $\lim\limits_{n \to \infty} \sum\limits_{k = 1}^{n} {\sqrt k \over n^{1.5}} = {2 \over 3}$

I am currently trying to prove: $\lim\limits_{n \to \infty} \sum\limits_{k = 1}^{n} {\sqrt k \over n^{1.5}} = {2 \over 3}$ I can easily squeeze the series between 0 and 1. I don't know many handy ...
-2
votes
5answers
145 views

Help finding the limit of this series $\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \cdots$

How can I go about finding the limit of $$\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \cdots = \sum_{k = 1}^{\infty} \frac{1}{2^{k+1}}?$$ Could I use the absolute value theorem? I have a ...
1
vote
2answers
47 views

Conditional convergence and Riemann's series theorem

There are tests to determine whether an integral or sum is convergent. There are test to determine whether an integral or sum is absolutely convergent. An integral or series is said to be $\mathbf ...
3
votes
2answers
61 views

Find the sum$\pmod{1000}$

Find $$1\cdot 2 - 2\cdot 3 + 3\cdot 4 - \cdots + 2015 \cdot 2016 \pmod{1000}$$ I first tried factoring, $$2(1 - 3 + 6 - 10 + \cdots + 2015 \cdot 1008)$$ I know that $\pmod{1000}$ is the last ...
-2
votes
1answer
30 views
4
votes
1answer
85 views

Find the natural boundary of $\sum_{n=1}^\infty \frac{z^n}{1-z^n}$

I'm asked to prove that the natural boundary of $\sum_{n=1}^\infty \frac{z^n}{1-z^n}$ is the unit circle. My try: First, use root test to show that the series converges for $|z|<1$. Then I have ...
1
vote
1answer
101 views

Is $(\frac 1{n^2 \sin n })$ convergent ? If so , what is the limit? [duplicate]

Is the sequence $\left(\dfrac 1{n^2 \sin n }\right)$ convergent ? If it does, with what limit?
0
votes
2answers
27 views

suppose $a_n>1$ $a_n$is non-decreasing and bounded. prove: $\sum_{n=-1}^{\infty}(1-\frac{a_n}{a_{n+1}})\frac{1}{\sqrt{a_{n+1}}}$ converges

suppose $a_n>1$, $\{a_n\}$ is non-decreasing and bounded. prove: $\sum_{n=-1}^{\infty}\left(1-\frac{a_n}{a_{n+1}}\right)\frac{1}{\sqrt{a_{n+1}}}$ converges I don't have any idea about how to ...
0
votes
0answers
16 views

Convergence a series with non-negative terms and it's relationship with geometric series [on hold]

Suppose $\sum_{n=1}^{\infty} a_n$ is a convergent series of non-negative terms. 1, Does there then exist a $q \in (0,1)$ and $k \in \mathbb N$ , such that $a_n \leq q^n$ for all $n > k$ ? 2, Is ...
1
vote
0answers
23 views

Periodicity of Newton's method approximations on a cubic polynomial

Bruckner & Bruckner, Elementary Real Analysis Let $f(x) = x^3 - 3x + 3$ Applying Newton's method to get $x_{n+1} = x_n -\frac{f(x)}{f'(x)} \ ,$ prove that for any positive integer $p$, there ...
0
votes
4answers
50 views

Proof of Convergence of a Sequence

Show that the sequence $\frac{n^2+1}{n^2+n}$ converges and its limit is $1$. However, I am finding it difficult to prove according to the rules that a converging sequence must obey, that is, sequence ...
13
votes
2answers
406 views

Are eigenvalues of the limit of a sequence of matrices limits of eigenvalue sequences?

Let $\{A_n\}\in \mathbb{R}^{m\times m}$ be a sequence of symmetric matrices such that $A_n\to A$ as $n\to \infty$, i.e. $\lim_{n\to \infty}a_{ij}(n)=a_{ij}\ \forall 1\le i,j\le m$ where ...
-3
votes
1answer
37 views

For what values of α the following serie converges?

For what values of α the following series converges? $\displaystyle\sum_{n=1}^{\infty} (\frac{1}{n}-\sin\frac{1}{n})^{\alpha}$ Help.. Thanks...
4
votes
1answer
33 views

Rationality of subseries

Let $(a_n)_n$ be a real sequence. Assume than $\sum_{n=1}^\infty a_n$ converges to a rational number. Does there exist a subseries which converges to an irrational constant? Assume now the opposite ...
0
votes
3answers
73 views

Proof the series is finite using following inequality

Let $$a_n=\frac1{\sqrt1}+\frac1{\sqrt 2}+\ldots +\frac1{\sqrt n}-2\sqrt n $$ For the task to prove that $$\tag1-2\le a_n\le -1 $$ I was given the hint $$\tag2\sqrt{k+1}-\sqrt k<\frac1{2\sqrt ...
3
votes
1answer
69 views

Limit of given expression

Let $\sum a_k=s$. I want to show that $$\lim\limits_{x\to 1^-}(1-x)\sum\limits_{k=1}^{\infty}\frac{ka_kx^k}{1-x^k}=s$$ where $x\in(0,1)$. Thanks for your helps.
1
vote
1answer
33 views

Proving the closed form for an infinite sum (related to Chebyshev polynomials)

How do I prove the following identity? For $y\not= 0$, we have $$ \sum_{n=0}^{\infty} \dfrac{1}{2y}\left( (x+y)^{n+1}-(x-y)^{n+1}\right) = \dfrac{1}{(x+y-1)(x-y-1)}. $$ I am trying to find the ...
2
votes
1answer
79 views

How to calculate the $n$-th member of sequence $a_{n+1}=\sqrt{y+a_{n}}$

I was searching for a smooth continuous concave function $$f:R^{+}\times R^{+}\to R^{+}$$ so that $$f(x+1,y)=\sqrt{y+f(x,y)}\quad\text{and}\quad f(0,y)=0.$$ But I couldn't find a general function, ...
2
votes
2answers
43 views

Converge series such that permuting the termes will change the limit.

I know that for a series that converge, if we permute the element of the sum, the series doesn't necessarily converge. For exemple $$\sum_{n=1}^\infty \frac{(-1)^n}{n}$$ converge but if we first sum ...