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

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-3
votes
4answers
100 views

How to prove that series $\frac{1}{n+1}$, as $n\to \infty$ is zero.

Can somebody explain how to prove that series $\frac{1}{n+1}$, as $n \to \infty$? I mean infinite series, not sequence, and I want to understand how to define the partial sum when n goes to infinity. ...
0
votes
0answers
27 views

How to Invert this formula

In this paper, under Stirling Numbers and their Asymptotics, the author takes equation (3.1): ...
0
votes
5answers
93 views

how to prove that $\lim\limits_{n\to\infty} \left( 1-\frac{1}{n} \right)^n = \frac{1}{e}$

I need to prove $$\lim\limits_{n\to\infty} \left( 1-\frac{1}{n} \right)^n = \frac{1}{e}$$ For now, I only know the e definition: $\lim\limits_{n\to\infty} \left ( 1+\frac{1}{n} \right)^n = e $, and I ...
0
votes
0answers
32 views

Invert this formula

How do I invert this to solve for $P(g_n(k)\cdot s)$? $$Z(s)=\left(1+\sum_{n>0}\prod_{k=1}^t \frac {(-1)^{f_n(k)}}{{f_n(k)}!}\left(\frac {P({g_n(k)}\cdot s)}{g_n(k)}\right)^{f_n(k)}\right)^{-1}$$ ...
1
vote
2answers
26 views

Prove that $\tan \left ( \sum_{k=1}^{n} \theta_k \right ) \geq \sum_{k=1}^{n} \tan (\theta_k)$

I'm trying to prove by induction that $$\tan \left ( \sum_{k=1}^{n} \theta_k \right ) \geq \sum_{k=1}^{n} \tan (\theta_k)$$ provided that $$\sum_{k=1}^{n} \theta_k < \frac{\pi}{2}$$ So in ...
3
votes
3answers
46 views

Expanding $\frac{2x^2}{1+x^3}$ to series

So I was doing some series expansion problems and stumbled upon this one ( the problem is from Pauls Online Notes ) $$f(x) = \frac{2x^2}{1+x^3}$$ The actual solution to this problem uses a ...
0
votes
2answers
74 views

Two divergent series such that their product is convergent

I faced a series question it goes something like give an example of 2 divergent series such that when the 2 series are multiplied to each other, the new series becomes convergent, although it looks ...
-2
votes
0answers
23 views

Can anyone provide proofs of the following problems? [on hold]

Problem: Let $x$ and $k$ be real numbers such that $x\ge1$ and $k>0$. Prove the following statements. (a) $-\frac{1}{x}+\sum_{n=1}^{\infty}\frac{1}{(nk+x)^2}\geq0$ (b) ...
1
vote
1answer
18 views

Sequence uniform convergence but the derivatives are not.

Give a sequence $(f_n)_{n\in \mathbb{N}}$ of differentiable functions which uniformly converge to $0$, but for which the seqeunce $(f_n')_{n\in \mathbb{N}}$ of the derivatives isn't even pointwise ...
5
votes
5answers
125 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 ...
-3
votes
2answers
68 views

Express the number $4$ and $5$ and $6$ and $7$ and $8$ [on hold]

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
22 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
41 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
72 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
38 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
36 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
383 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
63 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: ...
8
votes
2answers
115 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 ...
5
votes
1answer
84 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
78 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
54 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
153 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
38 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
87 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? ...
-4
votes
2answers
107 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
69 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
48 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
27 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
76 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
97 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
71 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
64 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
31 views
4
votes
1answer
88 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
104 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
28 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 ...