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

learn more… | top users | synonyms (5)

5
votes
2answers
223 views

series involving Catalan and Zeta

I ran across another challenging and interesting series, and I am wondering if someone could shed some light on how to evaluate it. $$ ...
3
votes
2answers
198 views

If $\sum a_n$ converges, then $\sum \sqrt{a_na_{n+1}}$ converges

Prove that if the positve term series $\sum^{\infty}_{n=1}a_n$ is convergent, also $\sum^{\infty}_{n=1}\sqrt{a_na_{n+1}}$ is convergent. Prove that if the positive term series ...
4
votes
3answers
411 views

Summation of a finite series involving permutations.

$$\large \sum_{i = 2}^{25}P(i,2)$$ $P$ stands for "permutations".
1
vote
1answer
77 views

Polynomial Formula like Infinite Sum with non-natural index

By polynomial formula $$(\sum_{i\in m} x_i)^n=\sum_{\substack{j_i \in \mathbb{Z}^+ \\ \sum j_i=n}}\left(\begin{array}{c} n\\ j_{0},\ldots , j_{m-1} \end{array}\right)\prod_{i \in m} x_i^{j_i}$$ where ...
5
votes
3answers
246 views

Compute $\lim_{n\to\infty} \sum_{k=1}^{n} \left(\frac{k}{n^2}\right)^{\left(\frac{k}{n^2} + 1\right)}$

Compute the limit $$\lim_{n\to\infty} \sum_{k=1}^{n} \left(\frac{k}{n^2}\right)^{\left(\frac{k}{n^2} + 1\right)}$$ At a first look, I only thought of Riemann sums, but I don't see how I may apply ...
3
votes
3answers
723 views

Inequality of finite harmonic series

I'm asked to prove that for $n\in \mathbb{N}$ $$\frac{1}{1} + \frac{1}{2} +\cdots+\frac{1}{n} \geq 1 + \frac{n}{2}$$ by induction. I've got a feeling that the problem isn't right (since it isn't true ...
1
vote
1answer
537 views

Example that a measurable function $f$ on $[1,\infty )$ can be integrable when $\sum _{n=1}^{\infty }\int_{n}^{n+1}f$ diverges.

I am seeking help in my attempt to formulate a proof to disprove the following. For a measurable function $f$ on $[1,\infty )$ which is bounded on bounded sets, define $a_n= \int_{n}^{n+1}f$ for each ...
1
vote
1answer
67 views

Summation of Modulo Sequences

I came across through simulation that multiplying and adding certain modulo sequences yield equal results. Consider the following two sequences \begin{align} g_0[k] &= \sum_{n=0}^{N-1} \left< ...
5
votes
1answer
717 views

When does the summation of a quotient equal the quotient of summations?

That is, under what conditions would $$ \sum_{i = 1}^n \frac{a_i}{b_i}= \frac{\sum_{i = 1}^n a_i}{\sum_{i = 1}^n b_i} $$ be true? What about for infinite summations, i.e. when $n \rightarrow ...
3
votes
3answers
366 views

How to get the N-th word in a sequence

Suppose I have an alphabet (e.g. consisting of ABCDEF) and a lexicographic order is defined i.e. A -> B ... -> F -> AA -> AB .. -> AF -> BA -> BB -> ... -> BF ... -> FF -> AAA -> ... Is there a ...
0
votes
2answers
244 views

Formalizing the idea of “algorithm”

I have encountered several times, while doing mathematics, the following situation: We have a finite "sequence" $\left(a_{n}\right)_{n\in\left\{ 1,\ldots,p\right\} }$ of some objects that has the ...
7
votes
1answer
270 views

If a sequence satisfies $\lim\limits_{n\to\infty}|a_{n+1} - a_n|=0$ then the set of its limit points is connected

Prove that if a sequence satisfies $\lim\limits_{n\to\infty}|a_{n+1} - a_n|=0$ then the set of its limit points is connected. My professor once mentioned a proposition likewise but I cannot find ...
3
votes
1answer
433 views

Counting Hexagons in Triangle Grid Recurrence?

(This is from a long finished programming competition) Consider a triangle grid with side N. How many hexagons can fit into it? This diagram shows N = 4: I need a recurrence for it: I tried the ...
5
votes
2answers
300 views

Convergence of a sequence of real numbers

Let $\alpha, \gamma$ be real numbers such that $0<\alpha<1$ and $\gamma>0$. Consider the sequence of real numbers given by $$ \begin{cases} x_0\ne 0&\\ ...
3
votes
1answer
450 views

Calculating digamma and trigamma functions

What is the best method to calculate the value of digamma and trigamma functions? Wikipedia suggests using recurrence relations $\psi_0(x+1) = \psi_0(x) + 1/x$, $\quad\psi_1(x+1) = \psi_1(x) - 1/x^2$ ...
3
votes
4answers
173 views

Does $\alpha_n = \left(\frac{1}{n}\right)^{\frac{1}{n}}$ converge to $1$?

Does the sequence $\alpha_n = \left(\frac{1}{n}\right)^{\frac{1}{n}}$ converge to $1$? If yes, how can i show that? I tried various simple methods unsuccessfully.
2
votes
3answers
119 views

Show this summation diverges

I would like to show that $$\sum \frac{1}{nn^{1/n}}$$ diverges, and I'm quite certain I will have to use the comparison test. I don't immediately see how that would be useful, though.
2
votes
4answers
130 views

Limits for sequences and the $e$ method

After doing a lot of homework, I've realized that I don't really get how to use the $\left(1+\frac{1}{n}\right)^n\to e$ method, which shows in how I can't solve indeterminate limits of the type ...
1
vote
1answer
172 views

Series - Bound by integrals

I have a series, $1^3 + 2^3 + 3^3 ... n^3$, and I want to find the upper and lower bound of this series using integrals. I know that for a series that is decreasing (such as $\frac{1}{1^2} + ...
2
votes
3answers
216 views

Find the sum $\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\dots+\frac{1}{99\cdot 100}$

Please help me calculate the following sum $$\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\dots+\frac{1}{99\cdot 100}$$
1
vote
2answers
96 views

Do two different metrics give the same limit?

Let's consider a set $X$ with two different metrics (distance function) $d_1, d_2$ on $X$. Is $\lim_{n\to\infty} d_1(x_n,x)=0 $ equivalent to $\lim_{n\to\infty} d_2(x_n,x)=0$? I mean, when we can ...
3
votes
1answer
154 views

Conditional sum and absolute sum

My professor in a Measure theory class mentioned that there is a difference between $$\sum_{n\in\mathbb{N}}a_n$$ and $$\sum_{n=0}^\infty a_n,$$ in the sense that the latter considers the order (taking ...
1
vote
3answers
62 views

More and more limits for sequences

So here goes a bit of homework: $$\lim_{n\to\infty}{\left(\frac{3n^2+2n+1}{3n^2-5}\right)^{\frac{n^2+2}{2n+1}}}$$ Well, this would trivially lead to: ...
2
votes
1answer
133 views

Finding $\pi$ through sine and cosine series expansions

I am working on a problem in Partha Mitra's book Observed Brain Dynamics (the problem was originally from Rudin's textbook Real and Complex Analysis, and appears on page 54 of Mitra's book). ...
12
votes
4answers
280 views

Calculation of a strange series

Is it possible to find an expression for: $$S(N)=\sum_{k=0}^{+\infty}\frac{1}{\sum_{n=0}^{N}k^n}?$$ For $N=1$ we have $$S(1) = \displaystyle\sum_{k=0}^{+\infty}\frac{1}{1 + k} = ...
1
vote
1answer
75 views

zeros of linear recurence sequences

Given a linear recurrence sequence $\{a_n\}_{n\geq 0}$, how to decide whethere there are infinitely many zeros, or there are only finitely many ones?
3
votes
2answers
87 views

Yet another limit to a sequence

so, I'm completely stuck with this limit: $$\lim_{n\to\infty}{\left(1+\frac{1}{n^2}\right)^n}$$ Which I can't even grasp how to start with, since I didn't understand the explanation of some method ...
1
vote
1answer
200 views

Need formula for sequence related to Lucas/Fibonacci numbers

I am trying to get a formula for the nth term of the following sequence: 2, 14, 22, 58, 266, 398, 1042, 4774, 7142, 18698, 85666, 128158, 335522,... It's not in OEIS and as far as I can tell ...
3
votes
1answer
90 views

On the limit of a sequence

My homework asks me to calculate (if it exists) the following limit: $$\lim_{n\to\infty}{\frac{(1+(-1)^n)^n}{n}}$$ My thinking is: $(-1)^n$ would, as we all know, oscillate between 1 and -1, meaning ...
21
votes
17answers
3k views

How can I get sequence $4,4,2,4,4,2,4,4,2\ldots$ into equation?

How can I write an equation that expresses the nth term of the sequence: $$4, 4, 2, 4, 4, 2, 4, 4, 2, 4, 4, 2,\ldots$$
3
votes
2answers
88 views

Calculate $\lim_{n\to\infty}[(1+x)(1+x^2)(1+x^4)\cdot\cdot\cdot(1+x^{2n})]$, $|x|<1$

Please help me solving $\lim_{n\to\infty}[(1+x)(1+x^2)(1+x^4)\cdot\cdot\cdot(1+x^{2n})]$, $|x|<1$
7
votes
2answers
256 views

A young limit $\lim_{n\to\infty} \frac{{(n+1)}^{n+1}}{n^n} - \frac{{n}^{n}}{{(n-1)}^{n-1}} =e$

These days I saw a really interesting limit as I was reading more information on Napier's constant here : http://mathworld.wolfram.com/e.html. It seems a pretty young limit since it appears under the ...
4
votes
1answer
408 views

Convergence of the Eisenstein series

Consider the complex semiplane $\mathcal H=\{z\in\mathbb C\,:\,\Im(z)>0\}$, and lets indicate with $\tau$ the elements of $\mathcal H$. Serre's book "A course in arithmetic" says that the serie of ...
1
vote
1answer
254 views

Uniform convergence problem for sine function

I want to show that $\sum^{\infty}_{n=1}\frac{\sin(nx)}n$ converges uniformly on $[a, 2\pi - a]$ for $0<a<2\pi$ Actually, I know $\sum^{\infty}_{n=1}\frac{\sin(nx)}n$ converges (by using ...
2
votes
1answer
681 views

Fubini theorem for sequences

I want to find a counter example This is the Fubini theorem for sequences: If $$\sum^{\infty}_{n=1}\sum^{\infty}_{m=1}|a_{mn}|<\infty,$$ then ...
1
vote
2answers
157 views

Difference of harmonic series

Proving convergence: $$\sum_{n=1}^\infty (-1)^{n-1}\frac1n$$ Just wanted to confirm if the reason they converge is due to the fact that for n= 1, 3, 5, ... we have a positive harmonic series and for ...
4
votes
2answers
2k views

Sequence Sum {1/2 + 1/4 + 1/6 +…} to infinite

I've been told, the following series converges: $$\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\ldots+\frac{1}{2k}+\ldots$$ I can't get my head around, how to prove this converges; any hints?
2
votes
0answers
76 views

how to show that a series converges without use of limits.

Just wanted to know if there is another method? Of the methods I know: Ratio test Comparison test Root test Integral test Limit comparison test All make use of limits. The reason why I am ...
4
votes
2answers
1k views

Convergent or divergent series examples

Suppose $\sum a_n$ is convergent. Is $\sum {{a_n} \over {1+|a_n|}}$ convergent or divergent?
2
votes
2answers
193 views

A question with the sequence $e_{n}=\left(1+\frac{1}{n}\right)^{n}$

Prove that $a$) the following sequence is increasing $$e_{n}=\left(1+\frac{1}{n}\right)^{n},\quad n\ge1;$$ $b$) the inequality below holds $$e_{n} \leq3,\quad n\ge1.$$
3
votes
3answers
124 views

Accuracy from approximating $\zeta(2)$ with a partial sum

This is for an introductory numerical analysis class. The answer shouldn't be too complicated, but if you have one, feel free to post it. Figure out what $n$ should be such that $$\sum_{k=n+1}^\infty ...
7
votes
1answer
366 views

If $\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$ then $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$

This is a question from the book Methods of Real Analysis by R. R. Goldberg. If $(s_n)$ is a sequence of real numbers and if $$\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$$ then prove that: ...
8
votes
2answers
1k views

Explicitly finding the sum of $\arctan(1/(n^2+n+1))$

This is from a GRE prep book, so I know the solution and process but I thought it was an interesting question: Explicitly evaluate $$\sum_{n=1}^{m}\arctan\left({\frac{1}{{n^2+n+1}}}\right)$$.
0
votes
3answers
68 views

Infinite series: When sum f(t) = sum f(t)g(t) ?

I was wandering if there are any theorems/ideas that would help me with the following situation (I came up with it myself and have been unable to find anything whatsoever). Say you have two ...
14
votes
3answers
714 views

Compute: $\sum_{n=1}^{\infty}\dfrac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)\cdot (2n+1)}$

Compute the sum: $$\sum_{n=1}^{\infty}\dfrac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)\cdot (2n+1)}$$ At the moment, I only know that it's convergent and this is not hard to see if you look at the ...
2
votes
1answer
453 views

Formula to calculate the sum of the series [duplicate]

Possible Duplicate: Counting subsets containing three consecutive elements (previously Summation over large values of nCr) Suppose there are n houses and I want to calculate the number of ...
2
votes
3answers
121 views

Does this number have an expression as square root or log of something

If this ends up being a ridiculous question I will delete it. Forgive me if this is ridiculous but this number has me stumped. $$1.52360679774998$$ The continued fraction calculator gives $1, 1, 1, ...
2
votes
3answers
180 views

Can you help me understand this definition for the limit of a sequence?

I'm reading the textbook "Calculus - Early Transcendentals" by Jon Rogawski for my Calculus III university course. I'm trying for the life of me to understand the wording of this definition, and I ...
3
votes
4answers
210 views

Why does the tail $a_N+a_{N+1}+a_{N+2}\ldots$ of convergent series $\sum a_n$ tend to $0$ as $N\to\infty$?

Let $a(n)>0$ for all $n \in \mathbb{N}$ be such that $\sum a(n)$ converges. Define $r(n):=\sum\limits_{k=n+1}^\infty a(k)$. The claim is $$\lim_{n \to \infty} r(n) = 0,$$ but I cannot see how to ...
1
vote
1answer
313 views

How to remove the denominator?

I have the following expression for $n>3$: $$\frac{5\cdot(n-1)\cdot[8\cdot\operatorname{Luc}(n) + 5\cdot\operatorname{Luc}(n-1)] + [4\cdot\operatorname{Luc}(n-1) - ...