For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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4
votes
1answer
178 views

The Series $-\frac{2}{5}+\frac{4}{6}-\frac{6}{7}+\frac{8}{8}-\frac{10}{9}-\cdots$

Stewart claims that this series is convergent, but Wolfram and I disagree. I looked at $$\lim\limits_{k\to\infty}\dfrac{(-1)^k (2k)}{4+k} $$ which is clearly not 0. Did I do something wrong?
1
vote
0answers
30 views

Prove the uniform convergence of series in a compact space

I have to prove that if $\sum\limits_{n=1}^\infty \frac{1}{|a_n|}$ converges, then $\sum\limits_{n=1}^\infty \frac{1}{x-a_n}$ converges absolutely and uniformly in any compact space that $\forall n \...
0
votes
2answers
44 views

Let $a_n\in A$ and $\lim_{n \to \infty}|a_n-a_{n+1}|=0$,then which is true?

Let $a_n\in A$ and $\lim_{n \to \infty}|a_n-a_{n+1}|=0$,then which is true? $1.$There exists $a\in A$ such that $a=\lim_{n \to \infty}a_n$ $2.$There exists $b\in \mathbb{R}$ such that $b=\lim_{n \to ...
3
votes
1answer
80 views

Convergence of $\sum x_k \log (x_k^{-1}) $ [closed]

Prove or disprove: If $\{x_k\} \subset (0,1)$, and $\sum x_k < \infty$, then $\sum x_k \log (x_k^{-1}) < \infty$. What if $\{x_k\}$ is monotone? Thanks a lot.
1
vote
0answers
38 views

Question about series and how the pattern idea works

Two Questions: When you are given: $1, 2, 3, .... , n$ How do you know that in the $...$ that it continues the $x_{n-1} + 1$ pattern? Is it the definition of series? Secondly: Do partial sums ...
3
votes
1answer
84 views

Prove that $\lim_{n\to\infty} \left( \frac{ g_n^{\gamma}}{\gamma^{g_n}} \right)^{2n} = \frac{e}{\gamma}$

Put $g_n = 1 + \frac{1}{2} + ... + \frac{1}{n} - \log(n)$. Prove that $$\lim_{n\to\infty} \left( \frac{ g_n^{\gamma}}{\gamma^{g_n}} \right)^{2n} = \frac{e}{\gamma}$$ I've tried this for a while now ...
1
vote
0answers
177 views

Simple finite series with reciprocal factorials

I'm trying to find the following sum: $$ \sum_{k=0}^n{1\over(n-k)!}{x^{k+2}\over k+2}. $$ The most obvious way is to differentiate wrt to $x$ leading to $$ \sum_{k=0}^n{1\over(n-k)!}{x^{k+1}}=\frac{1}...
1
vote
3answers
72 views

Is sequence limited and what is limit

I am stuck at one problem. So I have to check if sequence is convergent. $$\frac{2^x}{x!}$$ My thinking was to calculate limit and if limit exists it's convergent, but I am struggling with this: $$\...
-1
votes
1answer
37 views

How to derive this inequality

I learnt that for a standard normal random variable $Z$ and positive $x$, we have $$\mathbb P (Z > x) \geq \frac{x}{x^2+1} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} = \frac{1}{x+\frac{1}{x}} \frac{1}...
3
votes
1answer
57 views

The sequence $\sin \left({n\pi}\over 6\right)$ has the superior limit $L=1\dots$

I am studying limit points of a sequence now, and have some misunderstandings. Here's an exercises I have: The sequence $$\sin \left({n\pi}\over 6\right)$$ has the superior limit $L=1$and the inferior ...
6
votes
3answers
131 views

Is there any method to get a finite sum for $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$?

As we can see on Wikipedia, there are some algebraic methods that give us finite sums for the Grandi's series $$1-1+1-1+1-1+1-1+\cdots$$ Let $S$ be the sum of the Grandi's series. Then $S=(1-1)+(1-...
1
vote
2answers
51 views

Simple equivalent of the rest of the series $\sum\limits_n\frac1{n^3}$

Consider the converging series \begin{equation} \sum_{n\geqslant1}{\frac{1}{n^3}} \end{equation} I want to find an equivalent of the rest : \begin{equation} R_n=\sum_{k=n+1}^{\infty}{\frac{1}{k^3}} \...
2
votes
0answers
60 views

Time to buy a house without a mortgage equation!!

I am looking into a "real world" calculation to calculate the time taken for someone to buy their own home while they rent it. They do this by buying small pieces of the property every month, and ...
4
votes
2answers
208 views

Proof that $\sum\limits_{i=1}^n \cos \sqrt{i}$ is unbounded.

Please advice how to prove that $\sum\limits_{i=1}^n \cos \sqrt{i}$ is unbounded. By this I mean there exists no positive real $B$ such that for any natural $n$ $$-B <\sum\limits_{i=1}^n \cos \sqrt{...
1
vote
1answer
32 views

Is there a way to simplify $\sum_{j=0}^{n}C_{n}^{j}\sum_{i=0}^{m}\frac{j}{j+i}C_{m}^{i}$

Is there a way to simplify the following sequence? $$ \sum_{j=0}^{n}C_{n}^{j}\sum_{i=0}^{m}\frac{j}{j+i}C_{m}^{i} $$
4
votes
1answer
51 views

Sequences and limits

Let (an) be a sequence with an>0 for all natural numbers n. Assume that lim(an)=0. Show that the set of all numbers an has a maximum. That is, show that there is some number p, such that an <=ap. ...
2
votes
3answers
136 views

If a set $S\subset\mathbb R$ is not closed, does it contain a convergent sequence with a limit outside of $S$?

Suppose S is a subset of $\mathbb{R}$ and that S is not closed. Must it follow that there is a convergent sequence in S that converges to some l not in S?
7
votes
2answers
164 views

Evaluating a series of hypergeometric functions

I would like to prove (or disprove) the following statement: $$ \sum_{n=0}^\infty \left[\frac{{}_2{\rm F}_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};1\right)}{n!}\right] = \frac{\pi}{2} \left[ \sum_{...
1
vote
2answers
77 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
62 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
25 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
67 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
21 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 ...
1
vote
2answers
175 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
69 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
41 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 $\sum _{k=0}^{\infty}x^n=\frac{1}{1-x}$...
0
votes
2answers
43 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
65 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 ...
1
vote
1answer
302 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
5answers
4k 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 $\forall\...
-1
votes
1answer
58 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 $\lim (S_n) = \lim (S_{n+1}) = \lim (S_{n+2}) = s$?
1
vote
1answer
46 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 $\...
3
votes
3answers
89 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, ...
5
votes
8answers
179 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
222 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
48 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
77 views

Finding $a_n$ if $a_{n+1}=3a_n-2a_{n-1}$ with $a_1=2$ and $a_2=3$

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.
8
votes
4answers
806 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 \...
1
vote
2answers
86 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 $$ \sum_{...
2
votes
2answers
81 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
124 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
75 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
260 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 ...
7
votes
2answers
677 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
88 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
38 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 ...
1
vote
2answers
64 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
47 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
134 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
213 views

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 $$\sum_{k=0}...