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

learn more… | top users | synonyms (5)

0
votes
2answers
23 views

Help on Geometric Sequence Problem?

The sum of an infinite geometric series with first term a and common ratio r < 1 is given by $ S_n=a\cdot\dfrac{r^n-1}{r-1} $. The sum of a given infinite geometric series is $S_{\infty}=200 $ and ...
4
votes
2answers
57 views

If $g$ is continuous then $x^ng(x)$ converges on $[0,1)$

Suppose $g:[0,1]\to\mathbb R$ is a continuous function satisfying $g(1)=0$. Prove that the functions $f_n(x)=x^ng(x)$ converge uniformly on $[0,1]$. Hence or using Mean Value Theorem, prove that if ...
0
votes
0answers
36 views

Convergence of the complex series $\sum^{\infty}_{n=1}\epsilon_n\prod_{i=0}^{n-1}(z-w_i)$

I need to know this in order to prove something: Let $w_n$ ($n \geq 0$) be a sequence of complex numbers and let $\epsilon_n$ ($n \geq 1$) be another sequence of complex numbers such that for all $n ...
0
votes
2answers
48 views

limit of sequence of quotients of sequence that converges

Let $$\lim_{n\to \infty}x_n=a$$ Prove that if $$\lim_{n\to \infty}{x_{n+1}\over x_n}=L$$ so $$|L|\le1$$ .... I tried for a long time but i can't prove that. please give me just a hint? thanks
2
votes
1answer
59 views

Bound for the sum of a finite sequence

Consider ${\bf c} = (a,b) \in \mathbb{R}^2$ with $0< \|{\bf c}\| < 1.$ Let $n \in \mathbb{N} $ and define \begin{align*} F_{n}(k) & := \frac{ [a + x_{n}(k)]^2}{ [a + x_{n}(k)]^2 + [b + ...
1
vote
1answer
33 views

Evaluate $A=\lim_{n\to\infty} \int_a^b \operatorname{cotan}\left(\alpha x\right) \cos(n x) dx$.

Evaluate the limits $$A=\lim_{n\to\infty} \int_a^b \operatorname{cotan}\left(\alpha x\right) \cos(n x) dx$$ and $$B=\lim_{n\to\infty} \int_a^b \operatorname{cotan}\left(\alpha x\right) \sin(n x) dx$$ ...
2
votes
2answers
295 views

When is it allowed to take a constant out of a series?

When is it allowed to take a constant out of a series? Suppose we have a series $\sum ca_n$, when can we write it as $c\sum a_n$? It's pretty obvious when we know beforehand the series converges ...
4
votes
1answer
196 views

Sum the infinite series

How to solve this: \begin{equation*} \sum_{n=1}^{\infty }\left[ \frac{1\cdot 3\cdot 5\cdots \left( 2n-1\right) }{ 2\cdot 4\cdot 6\cdots 2n}\right] ^{3} \end{equation*} I can make the bracket thing, ...
1
vote
2answers
41 views

Proving the series doesn't converge: $\sum_{n=1}^{\infty}a_n$, $\lim_{n\to\infty}na_n=\infty$, $a_1=-1$

Let $\displaystyle\sum_{n=1}^{\infty}a_n$ and $\displaystyle\lim_{n\to\infty}na_n=\infty$ and $a_1=-1$. Prove the series does not converge. From the given that $a_1=-1$ we know that there has to ...
0
votes
2answers
61 views

Determine the radius of convergence of power series [on hold]

If I have the following functions : $$F(x)=\frac{2x}{1+x^2}$$ and $$G(x)=\frac{4x^2}{1+x^2}$$ and I want to determine the radius of convergence of power series , when I write each one of them as ...
-1
votes
1answer
25 views

Need help with this problem

If Sn denotes the sum to n terms of an A.P and p^2 Sp=pSp^2 for each p belongs to N then Sn is
0
votes
1answer
62 views

Maximise $y$ with respect to $x$ for $y=\prod_{k=1}^{\infty}(1-x^{-k})$

$$y=\prod_{k=1}^{\infty}(1-x^{-k})$$ I want to maximise this function. So far I have: $$\ln(y)=\sum_{k=1}^{\infty}\ln(1-x^{-k})$$ ...
0
votes
2answers
33 views

divergence of a sum

Let f(x) = xsin$(\frac{\pi}{2x})$, x$\in(0,1]$. Consider the sequence $(x_i) = \frac{1}{2n - (i-1)}$, i = 1,...,2n. Show that $\sum_{i=1}^{2n}|f(x_i)-f(x_{i-1})|$ diverges as n goes to infinity. ...
1
vote
3answers
61 views

Convergence of sequence $x_{n+1}=\sqrt{2x_n+3}$

Let $f:\Bbb R \rightarrow \Bbb R$ be the function $f(x)=\sqrt{2x+3}$. (a) Show that for all $x\in[1,3], f(x)\geq x$. You may use the intermediate value theorem if you like. Let $x_0=1$ ...
1
vote
1answer
23 views

$\lim_{n \to \infty} 2^{n + 2} \times (x^{2^n} - x^{2^{n+1}}) = 0$

For $x \in ]0,1[$, how can one see that: $$\lim_{n \to \infty} 2^{n + 2} \times (x^{2^n} - x^{2^{n+1}}) = 0$$ Thanks.
0
votes
0answers
43 views

Is there a closed-form expression for the following sum?

Is there a closed-form expression for the following sum: $$\large\sum_{\{n_i\}} \frac{x_i^{n_i}}{\prod_i (i!)^{n_i} n_i!}$$ where the sum runs over all combinations of $\{ n_{i=0,\dots,k} \}$ such ...
-1
votes
1answer
31 views

Infinite series to definite integral

Is there a method to transform the following infinite series to a definite integral? The problem is from my 1989 textbook on Calculus by Thomas and Finney, Ch 7., which focuses on methods of ...
4
votes
0answers
68 views

Help finding a closed form

I have the following function: $$\frac{2e^x}{e^{2x}+1+2x}=\sum_{n=0}^\infty \varepsilon_n\frac{x^n}{n!}$$ I would like to find a closed form for the $\varepsilon_k$. One thing that I do know is that ...
6
votes
4answers
132 views

What is the limit as k approaches infinity of $(k!)^{\frac{1}{k}}$

What is the value of $$\lim_{k\to\infty}(k!)^{\frac{1}{k}}?$$ One of my students concluded the limit was infinity – which I tend to agree with, but was unable to show that was the limit. We ...
3
votes
2answers
94 views

Convergence of $\sum \frac{a_n}{1+a_n}$ when $\sum a_n$ and $\sum a_n^2$ converges.

Suppose $a_n$ are real numbers and $\sum a_n$ and $\sum a_n^2$ converges. How would one go about showing that $\sum \frac{a_n}{1+a_n}$ converges? ($a_n \neq -1$ for every $n$)
2
votes
4answers
48 views

Finding what $p$ does the series: $\sum_{n=1}^{\infty}\frac{\ln n}{n^p}$ converges

For what $p$ does the series: $\displaystyle\sum_{n=1}^{\infty}\frac{\ln n}{n^p}$ converges? My attempt: I wanted to use the limit comparison test and compare it with $\frac 1 {n^p}$ but it ...
1
vote
1answer
27 views

Number of apples in a basket riddle

You have six baskets with apples - 10,12,15,20,22,25 (this is how many apples there were in them - 10 in first, 12 in second..). Some of the apples are red and some are green. After one basket was ...
1
vote
1answer
50 views

What is the sum of this series involving factorial in denominator?

$$1 + \frac{1^2 + 2^2}{2!} + \frac{{1}^2 + {2}^2 + 3^2}{3!} + \cdots$$ I can't figure out how to do summations which involve a factorial term in the denominator. Please help. This is a past year ...
0
votes
1answer
17 views

Formula for combinatorial series sum [duplicate]

As a part of one computer algorithm, I want to find sum for $$n+ \frac{n(n+1)}{2!} + \frac{n(n+1)(n+2)}{3!}+....+ \frac{n(n+1)(n+2)...(n+r-1)}{r!} $$. I looked at $$\frac1{(1-x)^n}$$. But it is ...
-2
votes
0answers
10 views

Notation for a sequence that is sorted in descending order

I have time-seriest data: l(t) I need to use a function that uses the 21 highest values in this time-series. Let's call the sorted index: j - j=1 is the maximum value of l(t); j=2 is the 2nd highest ...
1
vote
2answers
116 views

Convergence of averging series

If $\sum\limits_{n=1}^\infty a_n$ converges where $a_n>0,\ \forall n\in\mathcal N$, prove $\sum\limits_{n=1}^\infty \sqrt[n]{a_1a_2\cdots a_n}$ and ...
-3
votes
2answers
38 views

what's the limit of the following sequence? [closed]

$\displaystyle \frac{n^n}{3^nn!}$ Already tried to simplify the fraction but i can't solve the indetermination, please show me how to solve it step by step
1
vote
1answer
26 views

Problem related to means

I am confused on proving that If a, b, c are 3 numbers in harmonic progression Then ${(a^n+c^n) /2} >({(a+c)/2}) ^n$ I attempted like this... Since a, b, c are in hp so $(a+c) /2>b$ ...
1
vote
0answers
42 views

Strange sequence needed

There's no easy way to explain this, but please bear with me. I'll try to keep it slow and simple. I'm looking for a property that is related to the generalised pentagonal numbers (A001318 in the ...
0
votes
1answer
55 views

Where does this equation come from: $ (1+mx)^n = 1 + \sum_{n=1}^{\infty} {\binom{2n}{n} \over 4^n } x^n $

I have found the following problem here: https://brilliant.org/problems/intriguing-sum/?group=Km7yEIDGtHDa&ref_id=709399 In the solution a solver directly started with the equation given in the ...
2
votes
3answers
74 views

Converge or diverge $\sum^{\infty}_{n=2}\frac{4^n(n!)^2}{(2n-1)!}$

Show if the series converges or diverges.$$\sum^{\infty}_{n=2}\frac{4^n(n!)^2}{(2n-1)!}$$Can someone please help with proving this? (I think it converges)
0
votes
1answer
25 views

Relation between $A_{n-1}$ and $A_n$ in a sequence

Found this in a math textbook: If the relationship between terms $a_{n-1}$ and $a_n$ in a sequence is $3a_n = 2a_{n-1} +1$, then what is $a_{n+1}$? I am confused on how to solve this problem since ...
4
votes
4answers
548 views

Can this be shown: $\sqrt[3]{a\sqrt[3]{a\sqrt[3]{a\dots}}} = \sqrt a$?

$$\sqrt[3]{a\sqrt[3]{a\sqrt[3]{a\sqrt[3]{a\sqrt[3]{a\sqrt[3]{a\sqrt[3]{a\sqrt[3]{a\sqrt[3]{a\cdots}}}}}}}}}=\sqrt{a}$$ Just for fun. I would like to read the proof of this if it exists. Any ...
8
votes
1answer
136 views

Expressing ${}_2F_1(a, b; c; z)^2$ as a single series

Is there a way to express $${}_2F_1\bigg(\frac{1}{12}, \frac{5}{12}; \frac{1}{2}; z\bigg)^2$$ as a single series a la Clausen? Note that Clausen's identity is not applicable here.
0
votes
1answer
50 views

How can I show $| \sum_{k=1}^n e^{ik}|$ is bounded?

I know that we can write $ \sum_{k=1}^n e^{ik} = \frac{e^{i(n+1)} -1}{e^i - 1}$ But I am unsure how to proceed with showing there's some $M \in \mathbb{R}$ where $\forall n \in \mathbb{N} \space ...
1
vote
1answer
25 views

Finitude of an alternative of the Look and Say serie?

Consider the following alternative of the Look and Say sequence (OEIS A005150): $u_0=N>0$ $u_{n+1}$ is the number of 1s then the number of 2s then ... then the number of 9s in $u_n$. Example : ...
-1
votes
0answers
37 views

Convergent series and real numbers [closed]

Prove that every decimal representing a positive real number can be expressed as a convergent series. Any ideas?
6
votes
1answer
60 views

Prove that if $\sum a_n$ converges absolutely, then $\sum a_{2n}$ converges.

In posting this question, I noticed a lot of 'similar' threads pop up, but felt that they required a fundamentally different approach. If any of you feel differently, please feel free to vote this ...
0
votes
1answer
21 views

Consecutive terms which are all prime numbers but are also in AP

Let $a_1,a_2,a_3,\cdots$ be in AP with a common difference which is not a multiple of $3$.The maximum number of consecutive terms which are in AP and are also prime numbers is? I thought the answer ...
4
votes
3answers
138 views

To prove $\prod\limits_{n=1}^\infty\cos\frac{x}{2^n}=\frac{\sin x}{x},x\neq0$

Prove $$\prod_{n=1}^\infty\cos\frac{x}{2^n}=\frac{\sin x}{x},x\neq0$$ This equation may be famous, but I have no idea how to start. I suppose it is related to another eqution: (Euler)And how can I ...
0
votes
2answers
39 views

Generalized formula for sum of products.

Q:The sum of all possible products of the first n natural numbers taken two by two is? I did not understand the question as it is.What exactly is being asked?I'd really appreciate an answer ...
0
votes
1answer
27 views

Find $\sup_{x\in(0,\alpha)} \frac{n x^2}{x^3+n^3}$.

Find $$\sup_{x\in(0,\alpha)} \frac{n x^2}{x^3+n^3}$$ where $\alpha>0$ and $n\in\mathbb N$. I found $\sup$ doing the derivative. Is there an alternative way (without derivative)? Thank you very ...
1
vote
0answers
64 views

Sum of geometric series

Let's say I have the series: $1+(x+1)+(x+1)^2….$ if $|x+1|<1$, what is the sum of infinite geometric series? This is my thinking: I have the formula $S= a \dfrac{1-r^n}{1-r}$ Now we know that ...
1
vote
1answer
42 views

Number of the term

If the sum of n terms in AP is $3(n^2)+5$.What is the number of the term which equals $159$? My attempt: $3(n)^2-3(n-1)^2=159$.I got $n=27$ but the answer given is $21$.
0
votes
1answer
28 views

Finding the number of sequences in an arithmetic sequence. Exponential equation (?)

I have the value of sequence 1 (a1) and the difference between each sequence (d). a1 = 1 and d = 7. I have the sum of the arithmetric sequence (Sn) which is 1350. The task is to find the number of ...
1
vote
1answer
52 views

Converge the sequence $\left(\left(1+\frac{1}{n}\right) \left(1+\frac{2}{n}\right)\cdots\left(1+\frac{n}{n}\right)\right)^{1/n}$ [duplicate]

Does this series converge or diverge? $$ \left[\prod_{k=1}^{n}\left(1+\frac{k}{n}\right)\right]^{\frac{1}{n}} $$
1
vote
1answer
59 views

Find the Laurent expansion for $f(z)=\frac{\exp{1/z^2}}{z-1}$ about $z=0$.

Find the Laurent expansion for $f(z)=\frac{\exp{(1/z^2)}}{z-1}$ about $z=0$. I was able to determine the series for each of the factors. We have ...
1
vote
2answers
74 views

Comparing series

Can anyone explain why if I compare the coefficient of $x^{n}$ of the equation $$\sum_{k=0}^{\infty}a(n)x^n= \frac{1}{1-x}-\frac{x}{1-x^3}+\frac{x^2}{1-x^5}-\frac{x^3}{1-x^7}+...$$ I can get ...
2
votes
0answers
61 views

Why should we care about the root test?

In an introductory calculus class, students typically learn about both the ratio and root test for determining convergence of series. However, the ratio test seems sufficient enough for determining ...
4
votes
2answers
139 views

Nature of the serie $\sum\prod_{k=2}^n (2-e^{\frac{1}{k}})$

I'd like to determine the nature of the following serie : $$\sum_{n\ge 2}\prod_{k=2}^n (2-e^{\frac{1}{k}})$$ Let $u_n = \prod_{k=2}^n (2-e^{\frac{1}{k}})$. So I "have": $$\begin{aligned} \ln(u_n) ...