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

learn more… | top users | synonyms (5)

1
vote
2answers
43 views

Determine whether or not $\sum_{k=1}^{\infty} \frac{1}{k- \mathrm{e}^{-k}}$ converges.

I have the following so far Let $a_k = \frac{1}{k- e^{-k}}$. Now, $\lim_{k \to \infty}a_k=0 \implies$ $\sum a_k$ can either converge or diverege. We must thus do further tests to determine whether ...
0
votes
1answer
52 views

Why ${1\over n+1}+ \cdot\cdot\cdot +{1\over 2n}>{n\over 2n} $?

Can you please explain me why is : ${1\over n+1}+ \cdot\cdot\cdot +{1\over 2n}>{n\over 2n} ={1\over 2}$? Thank you very much.
0
votes
3answers
17 views

Expanding the series …

Here we have such a sequence $x_n$. The thing that I do not understand is the following: where does the right side of this equality come from, how is it formed ? Can you please show me the operation ...
2
votes
1answer
45 views

Suppose the limit of $f(z)$ as $z$ approaches $z_0$, exists and call it $w_0$. Suppose a sequence $(a_n)$ converges to $z_0$. Does $f(a_n)$ converge

Suppose the limit of $f(z)$ as $z$ approaches $z_0$, exists and call it $w_0$. Suppose a sequence $(a_n)$ converges to $z_0$. Does $f(a_n)$ converge to $w_0$ and $n \rightarrow \infty$? I would say ...
2
votes
2answers
16 views

In $(]0,1], d_{\mid.\mid})$ why $(\frac{1}{1+n})_{n\in \mathbb{N}}$ do not converge?

Can someone tell me why this sequence do not converge ? First, I know that is a Cauchy's sequence. Then, the fact is that the sequence converges to $0$ when $n \rightarrow \infty$. Thanks in ...
0
votes
0answers
12 views

Can Banach fixed-point theorem be generalized to the case where each term depends on multiple previous terms in recurrence sequence

Banach fixed-point theorem http://en.wikipedia.org/wiki/Banach_fixed-point_theorem I'm wondering if the theorem still applies if the recurrence relation is, for example, ...
1
vote
2answers
46 views

Accumulation points and closed sets

Denote by $F$ the set of all accumulation points of $(x_{n})$. We define an accumulation point $x \in \mathbb{R}$ if there exists a subsequence $(x_{n_{k}})$ of $(x_{n})$ (being the latter a bounded ...
0
votes
1answer
28 views

Subsequence converging to inf of sup

Let $(x_n)$ be a bounded sequence, and for each $n\in\mathbb{N}$ let $s_n=\sup\{x_k:k\geq n\}$ and $S=\inf\{s_n\}$. I need to show that there exists a subsequence of $(x_n)$ that converges to $S$. I ...
0
votes
2answers
32 views

Write $\sum_{k=0}^{\infty} \cos(k \pi / 6)$ in form $a+bi$

I have to write $\sum_{k=0}^{\infty} \cos(k \pi / 6)$ in form: $a+bi$. I think I should consider $z=\cos(k \pi / 6)+i\sin(k \pi / 6)$ and also use the fact that $\sum ...
0
votes
2answers
36 views

How to estimate $\sum_{n=0}^\infty (\log\log2)^n/n! $ from below?

How to show that the following inequality holds: $$\sum_{n=0}^\infty \frac{(\log\log2)^n}{n!}>\frac 35$$ Is it possible to prove this using induction?
0
votes
2answers
43 views

Find the limit of a sequence $(\frac{z^n}{n!})_{n=1}^{\infty}$

I have to find the limit of a sequence $(\frac{z^n}{n!})_{n=1}^{\infty}$ where $z$ is a complex number. I think it is zero, because we know that $\sum_{n=0}^{\infty} \frac{z^n}{n!}$ is finite. Is this ...
-5
votes
3answers
41 views

Non-Existence of Geometric Series Going to Infinity [closed]

According to this material, when analyzing the series $S = 1 + 2 + 4 + 8 + ...$ $S_n = \sum_{k=0}^n 2^k$ does not approach a specific value, so we say that the sum of the infinite geometric series ...
1
vote
0answers
68 views
+50

About a result concerning Mersenne primes

I want to verify the proof of this result Theorem: If $p>2$ is a prime and $$H_{p}=\frac{(\sqrt3+2)^{2^{p-1}}+1}{(2^{p}-1)(\sqrt 3+2)^{2^{p-2}}}$$ is a natural number then $2^{p}-1$ is a prime ...
1
vote
1answer
20 views

What will be nth term of the following sequence?

Let a, a+d, a+2d,...., be an A.P.If we eliminate every pth term, then what will be the new general value of nth term? For e.g. Let the A.P. be 2,5,8, 11 ,14,17,20, 23 ,26,29...[a=2, d=3] Now, if we ...
1
vote
6answers
222 views

How to prove a sequence does not converge?

I want to show that the sequence $$a_n=\frac{1}{n}+(-1)^n$$ does not converge to a limit. I know that if a sequence $\left( a_n\right)_{n \in \mathbb{N}}$ converges to a limit L, then ...
-2
votes
1answer
44 views

If $\lim (S_n)=s$, does it follow that $\lim (S_{n+1}) = \lim (S_{n+2}) = s$?

I have proven $\lim (S_n)=s$, where $S_n$ is a sequence. Am I allowed to say $\lim (S_n) = \lim (S_{n+1}) = \lim (S_{n+2}) = s$?
0
votes
1answer
30 views

Discuss convergence and find sum of the Series

Show that the series $\sum_{n=1}^\infty \ln(1-\frac{{1}}{10^n})$ converges and find the sum in closed form if it is possible. Try:Clearly given series converges because if $0<a_n<1$ then ...
2
votes
3answers
55 views

Show that $\sum_{k=1}^\infty \frac{\ln{k}}{k^2}$ converges or diverges

Show that $\sum_{k=1}^\infty \frac{\ln{k}}{k^2}$ converges or diverges I know that I must use the Limit Comparison test and my instinct tells me that this series will converge. I cannot, however, ...
0
votes
0answers
26 views

Analysis .Find sequences? [closed]

Find sequences $a_n, b_n$ such that \begin{align*} &a_n=\frac{x_n}{y_n} 3^{1/2}, \quad a_n \rightarrow 1 \\ &b_n = \frac{z_n}{w_n}, \quad b_n \rightarrow 3^{\frac{1}{2}} \end{align*} where ...
5
votes
8answers
123 views

Calculate $1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)$

How can I calculate $1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)$? I know that $1+2+\cdots+n=\dfrac{n+1}{2}\dot\ n$. But what should I do next?
1
vote
1answer
170 views

Prove $\sum_{n=1}^\infty \sum_{m=2^{n+1}} ^\infty \frac{(-1)^mn}{m}$ converges and evaluate it.

Noting that the series inside is a part of the alternating harmonic series multiplied by $-n$, we get $\displaystyle \sum_{m=2^{n+1}} ^\infty \frac{(-1)^mn}{m}=\sum_{m=1} ^\infty \frac{(-1)^mn}{m} ...
1
vote
1answer
38 views

Evaluating $ \sum_{i=0}^{\infty}ir^i$ for $|r|\lt 1$ [duplicate]

I'm doing my mathematics homework and there's question which I'm pretty much unable to solve. I've literally tried every method but no results. It'd be great, if anybody could help me. Thanks (in ...
0
votes
3answers
55 views

finiding $a_n$ if $a_1=2,\; a_2=3,\; a_{n+1}=3a_n-2a_{n-1}$

Given $a_1=2,\; a_2=3,\; a_{n+1}=3a_n-2a_{n-1}$. How can i prove that: $a_n=2^{n-1}+1$ I Tried to isolate $a_n$ but it doesn't get me anywhere. Thanks.
2
votes
0answers
39 views

How to calculate alternating Euler sum [closed]

In How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$? get $$\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2=\frac{1}{12}(\pi^2\log2-4(\log 2)^3-9\zeta(3)),$$ Similar, how to evaluate the series, ...
0
votes
0answers
41 views

Prove probing to be permutation

So I have been taught that probe sequences (h(k, 0), h(k, 1), ... , h(k,m - 1)) are meant to be a permutation (0, 1, ... ,m - 1), but how does one prove that? I was asked this question in an ...
5
votes
3answers
127 views

Some integral representations of the Euler–Mascheroni constant

What kind of substitution should I use to obtain the following integrals? $$\begin{align} \int_0^1 \ln \ln \left(\frac{1}{x}\right)\,dx &=\int_0^\infty e^{-x} \ln x\,dx\tag1\\ &=\int_0^\infty ...
0
votes
2answers
40 views

How to decide about the convergence of $\sum(n\log n\log\log n)^{-1}$?

In Baby Rudin, Theorem 3.27 on page 61 reads the following: Suppose $a_1 \geq a_2 \geq a_3 \geq \cdots \geq 0$. Then the seires $\sum_{n=1}^\infty a_n$ converges if and only if the series $$ ...
2
votes
2answers
46 views

Proving $z\,\Gamma(z) = \Gamma(z+1)$ using the product formula

I am trying to show that $z\,\Gamma(z) = \Gamma(z+1)$ using the product formula: $$ \Gamma(z) = \frac{e^{-\gamma z}}{z} \prod\limits_{n=1}^\infty\left(1+\frac{z}{n}\right)^{-1}e^{z/n}$$ where $\gamma$ ...
2
votes
1answer
29 views

How should Theorem 3.22 in Baby Rudin be modified so as to yield Theorem 3.23 as a special case?

I'm reading Walter Rudin's PRINCIPLES OF MATHEMATICAL ANALYSIS, 3rd edition, and am at Theorem 3.22. Theorem 3.22: Let $\{a_n\}$ be a sequence of complex numbers. Then $\sum a_n$ converges if and ...
1
vote
1answer
18 views

convergence criteria of an infinite series

$\sum _{n=1}^{\infty }{\frac {1}{50}}\,{\frac { \left( -1 \right) ^{1+n }{\it a}\, \left( 10000\,\cos \left( tn \right) \epsilon\,\delta_{{ 1}}-10000\,\cos \left( \frac{1}{10}\,\sqrt {4201}t \right) ...
1
vote
5answers
203 views

Why are Maclaurin series useful if we can only use them for such a small range of numbers?

Okay, I am beginning to get how Maclaurin series work, but what I don't understand is why they are useful. Why would you want an infinite expansion for a series that works for such few values (only ...
6
votes
2answers
587 views

Beautiful problem on a progression

$\{x_n\}$ is a sequence defined as follows: $x_1=20,\quad x_2=14,\quad x_{n+2}=x_n - \frac{1}{x_{n+1}}$. Prove that $0$ is among the members of this sequence. Find its number. I tried some stuff ...
3
votes
2answers
72 views

Is the following series convergent $\sum_{n=1}^{\infty}\dfrac{2^n+3^n}{3^n+4^n}$

Is the following series convergent $\sum_{n=1}^{\infty}\dfrac{2^n+3^n}{3^n+4^n}$ I tried everything, nothing appears to work. can some one give an idea
2
votes
1answer
21 views

Series of Sequence which always diverges

Suppose {$a_n$} is a sequence with $a_n>0$. For each $k$ in $\Bbb{N}$, set $$b_k = \frac{1}{k} \sum_{n=1}^{k}a_n$$ then woud $\sum_{k=1}^{\infty}b_k$ always diverge? I want to use Converge ...
0
votes
2answers
40 views

Solving an unusual equation

I need to find a real number $n$ such that $n > 1$ and: $$ \sum_{k=1}^\infty \frac{2^k}{n^k} = \frac{n-1}{n} $$ Ideally, I'd find the minimum such $n$ (if more than one exists), but really, any ...
2
votes
0answers
41 views

Convergence of $\sum_{n=1}^{\infty} \frac{Sin^2(n)}{n}$ [duplicate]

Is following sum convergent? $$\sum_{n=1}^{\infty} \frac{Sin^2(n)}{n}$$ Integral test, Dirichlet test doesn't apply. Any idea !
-1
votes
1answer
33 views

sequences: finding a formula for a function 4, 1, 2, 1, 4

I am trying to find and prove an arithmatic formula for a function. my teacher gave us a list of properties that the function meets. by using substitution on the properties given, i was able to find a ...
6
votes
2answers
130 views
+50

Closed-form of $\sum_{k=0}^{\infty} \frac{k^a\,b^k}{k!}$

While working on this question I think I've found a closed-form expression for the following series, but I don't know how to prove it. Let $a \in \mathbb{N}$ and $b \in \mathbb{R}$. Then ...
0
votes
2answers
25 views

$|\alpha _{n}| \le c_n$, then $\sum{\alpha _{n}}$ converges absolutely

Can you help me understand the part of the proof. Thanks Let $\sum{c_n}$ be a series of real numbers $\ge 0$ which converges. If $|\alpha _{n}| \le c_n$ for all n, then the series $\sum{\alpha _{n}}$ ...
8
votes
1answer
399 views

It is easy to show that $S_m=\sum_{n=1}^\infty \frac{n}{2^n + m}$ converges for any natural$\ m$, but what is its value?

In fact the series would converge even if$\ m$ were not natural, I just wanted to state that it is natural in my case. I have found the partial sum formula of$\ S_0$,$\displaystyle \sum_{n=1}^k ...
1
vote
1answer
35 views

Can anyone please clarify for me certain points I have in the proof of Theorem 3.17 in Baby Rudin?

I'm currently studying, or revising, Walter Rudin's Principles of Mathematical Analysis, 3rd eition, and I'm stuck with the proof of Theorem 3.17. Here's the statement of the Theorem: Let ...
0
votes
2answers
34 views

Questions Related to Sequences + Sums?

Let $a_n$ be the $n$th sequence 1, 2 , 2 , 3 , 3 , 3 , 4 , 4 , 4 , 4 , 5 , 5 , 5 , 5 , 5, . . . . . . . constructed by including the integer $k$ exactly $k$ time. Show that $a_n$ $=$ $\lfloor ...
5
votes
1answer
92 views
+50

Concerning the classical normalized Eisenstein series

Earlier I asked this question. As of today, it has not been answered. Yet still, I have a follow-up question: In general, how does one express $E_4(\tau)$ and $E_6(\tau)$ in closed form for special ...
3
votes
2answers
74 views

Closed-form of the sequence ${_2F_1}\left(\begin{array}c\tfrac12,-n\\\tfrac32\end{array}\middle|\,\frac{1}{2}\right)$

Is there a closed-form of the following sequence? $$a_n={_2F_1}\left(\begin{array}c\tfrac12,-n\\\tfrac32\end{array}\middle|\,\frac{1}{2}\right),$$ where $_2F_1$ is the hypergeometric function and $n ...
2
votes
0answers
65 views

The 6-order Euler sums

How to calculate the value of the series $$\sum\limits_{n = 1}^\infty {\frac{1}{{{n^4}}}\left( {\sum\limits_{k = 1}^n {\frac{{{{\left( { - 1} \right)}^{k - 1}}}}{{{k^2}}}} } \right)} ...
4
votes
4answers
221 views

Is there a sequence of strings where there is always an element between any two other elements?

Can I define a way of sorting strings such that for any two strings X and Z, I can find another one Y such that X < Y < Z ? This is clearly not the case for alphanumeric sorting: there is no ...
0
votes
1answer
49 views

Prove that all subsequential limits are contained within a closed interval

Let $a, b$ be two real numbers such that $a < b$, and suppose that $(s_n)_{n=1}^\infty$ is a sequence such that $\forall\,\, n\,\, a \leq s_n \leq b$. Prove that all subsequential limits are ...
5
votes
1answer
86 views

Proving a strange identity

Numerically, it would seem the following identity holds true: $$\frac{6}{7}=\lim_{n\to\infty}\sqrt[n]{\sum_{k=3}^\infty{\left(k-\sum_{j=1}^{k}\frac{1}{j}\right)^{-n}}}$$ Down below I have proven ...
0
votes
2answers
28 views

How to state convergence through limit comparison test?

I am able to show convergence of the following series through the root test but am trying to practice the limit comparison test and can't figure out how to do it that way. $$\sum_{n=1}^\infty ...
1
vote
1answer
32 views

Upper bound for an alternating lacunary series

I want to estimate an upper bound for the following alternating lacunary series $$\sum_{l=1}^\infty (-1)^{l+1}x^{2^l}$$ when $x$ is close to 1. In particular I want to show $$\limsup_{x\rightarrow ...