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

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2
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3answers
67 views

Use $\sum_{n=0}^{\infty} \frac {2n +1} {2^n} = 6 $ to show $\sum_{n=0}^{\infty} \frac {2n +1} {2^n} i^n = \frac 4 {25} + i \frac {22} {25}$.

I've shown that the series $$\sum_{n=0}^{\infty} \frac {2n +1} {2^n}$$ converges to $6$ using elementary series operations. However how can I use this to show that $$\sum_{n=0}^{\infty} \frac {2n +1} ...
1
vote
2answers
148 views

Difficult infinite trigonometric series

Evaluate the sum of the following infinite series. $$\left(\sin{\frac{\pi}{3}}\right) + \left(\frac{1}{2}\sin{\frac{2\pi}{3}}\right) + \left(\frac{1}{3}\sin{\frac{3\pi}{3}}\right) + \ldots$$
0
votes
1answer
32 views

Let $f(z) =: \sum^{\infty}_{n=0} (2n+1)z^n$. Find the power series representation of $g(z):=f(z)-zf(z)$ and show $g(z) = \frac {1+z} {1-z}$.

Let $f(z) =: \sum^{\infty}_{n=0} (2n+1)z^n$. I've verified that this power series has convergence radius $R=1$ be considering $\lim_{n \rightarrow \infty} |a_n|/|a_{n+1}|$ However I must find the ...
3
votes
1answer
59 views

Bound for a geometric sum

If $a>1$ and $0<x<1$ are fixed then there exist constants $c,c'$ such that $$ c\sum_{n=0}^N a^{nx} \leq (\sum_{n=0}^N a^n)^x \leq c' \sum_{n=0}^N a^{nx}$$ for all $N$, as one can see by ...
1
vote
3answers
124 views

Guess the closed form on the following sequence?

any help would be appreciated, have no idea where to start $u_1 = 2/3$ and $u_{k+1}$ such that: $$u_k + \frac{1}{(k+2)(k+3)}$$ for all, k are natural numbers guess a general formula (i.e the ...
0
votes
1answer
82 views

Uniform convergence of sequence of partial sums.Help please

Show that each sequence of partial sums and its derivative converges uniformly on their respective intervals. a) $$ S_n(x)=\sum_{k=1}^n \frac{1}{(1 + nx)^2}, x \in (0,\infty)$$ $$ ...
2
votes
0answers
36 views

Series of Real Numbers

Let $a_{k}$ be a sequence of real numbers such that $0 \lt a_{k+1}\le a_{k}$ for all k. If $\alpha\ge2$, is it true that $$\sum_{k=0}^{+\infty}(a_{k}-a_{k+1})^{\frac{1}{\alpha}}$$ converges?
1
vote
2answers
119 views

Show that each of the series converges.Help Please

Show that each of the series converges on their respective domains. a) $$\sum_{n=1}^\infty \frac{1}{(1 + nx)^2}, x \in (0,\infty)$$ b)$$\sum_{n=1}^\infty e^{-nx}, x \in(0,\infty)$$ For the first ...
1
vote
2answers
42 views

$a_3$, $a_5$ and $a_0$ terms are required?

We have an arithmetic sequence $a_n>0$ and it's increasing. and we've two systems of equations: $a_4=15$, $m+d=21$ whereas $m=lcm(a_3,a_5)$, $d=\gcd(a_3,a_5)$. What are the values of $a_3$, ...
0
votes
0answers
42 views

Find the interval of convergence for $\sum\limits_{n=0}^\infty\frac{(-1)^n}{n+1}\cdot x^{2n+2}$

I have a power series $$\sum\limits_{n=0}^\infty\frac{(-1)^n}{n+1}\cdot x^{2n+2}$$ and I need to find the interval of convergence. I'm not sure if I did this correctly. I said the interval of ...
0
votes
1answer
29 views

Finding radius of convergence

I have gotten the problem almost solved, but I'm hung up on how to solve this inequality: $$|x|/|2x+1|<1 $$ I could move the denominator to the right side of the equation: $$|x|<|2x+1|$$ But ...
1
vote
1answer
26 views

Can someone check my proof about uniform convergence of a series.Please

Suppose that $\sum |a_k| < \infty$, show that $$\int_0^1 \left(\sum_{k=0}^\infty a_kx^k\right)dx = \sum_{k=0}^\infty \frac{a_k}{k+1}.$$ I know I have to use the theorem that says, If $f_k$ is ...
1
vote
2answers
40 views

Uniform convergence of $\sum_{k=1}^\infty \frac{1}{k^{1+x}}$

Problem: Show that $\sum_{k=1}^\infty \frac{1}{k^{1+x}}$ converges uniformly on $[a,\infty)$ for any $a > 0$, but does not converge uniformly on $(0,\infty)$. what I have done: Let $f_n = ...
1
vote
1answer
50 views

series convergence test with parameter

As part of a bigger proof I reached the series $\sum {1 \over n^\alpha}$. $\alpha \in \mathbb{R}$ Obviously, the convergence depends on the value of $\alpha$. I already know the harmonic series ...
0
votes
0answers
13 views

Prove that $p_{k +1}$ = $P_0$$(\frac{b}{d})$$^{k+1}$ doesn't converge by using partial sum of geometric series

Prove that $p_{k +1}$ = $P_0$$(\frac{b}{d})$$^{k+1}$ does not converge by using the partial sum of the geometric series if the conditions are not met. I know that the condition is d > b where they ...
0
votes
1answer
46 views

Reducing a double series into a single series

References told me that a double series can be reduced into a single series by change of indices. Consider $\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} a_{n,m}$, by letting $m=p, n=p-q$, the series can be ...
0
votes
1answer
38 views

Convergence of Infinite Seriess

$$ \sum_{n=1}^{\infty} \frac{\sqrt{n+1}-\sqrt{n-1}}{n} $$ According to Mathematica, this series converges. I can't figure out how to prove this, however. If I split the series apart, I get two ...
0
votes
1answer
37 views

Find the interval of convergence of the series

Assuming that $x$ is fixed, I wrote: And then I used the Ratio test: So I get the interval of $1$ $<$ $x$ $<$ $3$. However, my question is: how can $sin$ be convergent at all? I mean, ...
1
vote
2answers
80 views

General formula for 1, 2, 9, 48, 300, 2160, …

What is the general formula for this sequence $$1, 2, 9, 48, 300, 2160, ...?$$
1
vote
2answers
35 views

Maximum value of a sequence

Find maximum value of the sequence $\left\{\frac{9^n}{n!}\right\}_{n\in\mathbb N}$ . As n tends to infinity the limit of the sequence goes to zero. And initially the value of the n-th ...
0
votes
3answers
41 views

Determine whether following series converges::

My thoughts: When n approaches infinity so given series diverges by Comparison test because $\frac{1}{n}$ diverges. Firstly, I wonder if I am correct on this one. And would there be another, ...
6
votes
1answer
428 views

On the possible values of $\sum\varepsilon_na_n$, where $\varepsilon_n=\pm1$ (i.e., changing signs of the original series)

I have used the following result in an answer on this site. Suppose that $a_n>0$ are positive real numbers such that $\sum\limits_{n=1}^\infty a_n=+\infty$ and $\lim\limits_{n\to\infty} ...
4
votes
2answers
56 views

Determine whether following series converges:

$$ \sum_{n=1}^\infty \frac{\sqrt{n}}{\sqrt{n^3}-i} $$ I determined that series diverges, because it's less than $\frac{1}{n}$ (I assumed that $i$ has no influence here) and $\frac{1}{n}$ diverges, ...
3
votes
2answers
122 views

Boundedness and Cauchy Sequence: Is a bounded sequence such that $\lim(a_{n+1}-a_n)=0$ necessarily Cauchy?

If I have a sequence {$a_n$} that has the property of $\lim(a_{n+1}-a_n)=0$, does that make it a Cauchy Sequence. I think it doesn't because $a_n = \sum_{k=1}^n \frac{1}{k}$ is a counter example. ...
2
votes
2answers
86 views

$a_1=3$ and $a_{n+1}=\frac{a_n}{2} + \frac{1}{a_n}$. Show that it monotonically decreases and find the limit.

What I've done so far: I have proved that this sequence is bounded below by 0, which is a very rough estimate. I know that the infimum is $\sqrt2$. Anyway, the question first asks me to prove that ...
2
votes
0answers
69 views

Proving a series is greater than zero

I wish to prove that an equation is greater than zero. Let ...
0
votes
1answer
35 views

Relation between coefficients of Maclaurin series and recurrence relations

Suppose $I_{2n}=\int_0^{\pi}x\sin^{2n}x \;dx$. Then one can obtain the following: $$I_0= \frac{\pi ^2}{2}, I_1= \frac{\pi ^2}{4}, I_2= \frac{3\pi ^2}{16}, I_3= \frac{5\pi ^2}{32},\ldots, I_n= ...
0
votes
5answers
291 views

How to create alternating series with happening every two terms

I'm looking for a technique for creating alternating negatives and positives in a series. Specifically: when n=1, the answer is +, n=2 is +, n=3 is -, n=4 is -... etc. I have every other part of the ...
0
votes
2answers
91 views

How to simplify the expression with sigma notation?

$$\sum\limits_{i=1}^n (n-i+1)(2i-1)^2= \frac{n(2n^3+4n^2+n-1)}{6}$$ how does this work? Could anybody show the details.
1
vote
4answers
53 views

$s_1 = 1$ and $s_{n+1}=(\frac{n}{n+1})s_n^2$ monotonically decreases?

Hi I came across this question on page 65 of Elementary Analysis by Kenneth A.Ross. I am given that $s_1 = 1$ and I need to show that $s_{n+1} = (\frac{n}{n+1}) s_n^2$ monotonically decreases. I'm ...
2
votes
2answers
95 views

If $\sum_{n=1}^\infty a_n$ converges and $\sum_{n=1}^\infty \frac{\sqrt a_n}{n^p}$ diverges, then p $\in$ {?}

Let {$a_n$} be a sequence of non-negative real numbers such that the series $$\sum_{n=1}^\infty a_n$$ is convergent. If p is a real number such that the series $$\sum_{n=1}^\infty \frac{\sqrt ...
0
votes
3answers
153 views

How to prove $\sum_{n=1}^\infty \frac{a_n}{1+a_n}$ converges iif $\sum_{n=1}^\infty\frac{a_n}{1-a_n}$ converges?

Could you please give me some hint how to prove this statement: If $0<a_n<1$ for each n, then $\sum_{n=1}^\infty \frac{a_n}{1+a_n}$ converges iif $\sum_{n=1}^\infty\frac{a_n}{1-a_n}$ ...
0
votes
0answers
46 views

Binomial series for $2^{n-1}\cos^n\vartheta$ and $2^{n-1}(-1)^{\frac{n}{2}}\sin^n\vartheta$

Can somebody confirm for me whether the following series are correct? $$2^{n-1}\cos^n\vartheta=\cos ...
1
vote
1answer
72 views

Interest Accumulation - Geometric Sequence

Hello I have just worked a question in which I get an answer different to the answer in my book. The question states: If a person deposits 500 at the end of each month for 20 years at an AER of ...
1
vote
1answer
45 views

Find the limit of $(a_k)_k$ where $a_k = \lim_{n\to\infty} \frac1n\sum_{m=1}^{kn} \mathrm{exp}(\frac12.\frac{m^2}{n^2})$

For $k \ge 1$, let $$a_k = \lim_{n\to\infty} \frac1n\sum_{m=1}^{kn} e^{\frac12.\frac{m^2}{n^2}}$$ Find the value of $$\lim_{k\to\infty} a_k$$
0
votes
1answer
234 views

Inverse of a lower triangular Toeplitz matrix vs. the matrix size

I am recently trying to find the inverse of the lower triangular Toeplitz matrix ($\mathbf{A}$), with some special elements: $$ \mathbf{A}_{M\times M}=\left[\begin{array}{ccccc} 1\\ -a_{1} & 1\\ ...
2
votes
1answer
55 views

The “trick” in the Herglotz trick

In How does the Herglotz trick work?, is explained as in "Proofs from THE BOOK" by Aigner and Ziegler, but the "trick" itself I found to be not so clear. The trick says: It follows from (4) ...
1
vote
2answers
92 views

a question about limit, I am struggling with this!

Suppose that {$a_n$}is a sequence of positive numbers.For each n which is a natural number,let $b_n$=($a_1+a_2+......a_n$)/n,prove that $\sum b_n$ diverges to $+\infty$. This question is my homework, ...
1
vote
1answer
60 views

Value of divergent series?

Let $\{a_n\}$ be a positive, convergent sequence. We consider the sequence of partial sum $\{s_n\}: s_n = \sum_{k=1}^n a_n$. Clearly $\{s_n\}$ is strictly increasing and therefore $\sum_{n=1}^\infty ...
1
vote
2answers
91 views

Absolute convergence does not implies convergence

Find a space and a series that converges absolutely but it does not converges. It is clear that the space can't complete or Banach.
3
votes
1answer
73 views

Sum as an integral

Recently I have encountered weird notation that I don't see into. When I have some infinite sum $$\sum_{n=1}^{\infty}f(n)$$ I would rewrite it without thinking to the integral form like this ...
1
vote
0answers
33 views

Prove that the limit of the function a sequence is the same as the function of the limit of the sequence [duplicate]

Assume that $\lim_{n\to \infty}a_n=a.$ Suppose that the function $f$ is continuous everywhere including at $a$. Form the sequence $(f(a_n))_{n=1}^{\infty}$. Prove that $\lim_{n\to ...
2
votes
4answers
48 views

Need help finding the limit of geometric serires

I'm learning some series tests in calculus and I can't completely figure this out. I know it's easier than i'm making it. Here's the question: Determine whether the geometric series is divergent ...
0
votes
2answers
40 views

Summation with non integer induces

All of the sums I've encountered so far have been functions $f:Z\rightarrow$ (any other set). Or in other words the sums are in the form of $\sum_{j\in Z\lor j\in N}^n a_j=a_0+a_1+...+a_n$. This ...
1
vote
2answers
34 views

Is the $\sum_{n=1}^{\infty} \frac{(2n+1)^{1/2}}{n^2}$ convergent or divergent?

For this question I am not really sure which test to use to determine this. I was thinking the comparison or limit comparison test but it doesn't seem to be working. I was wondering what the steps are ...
2
votes
3answers
94 views

How to show a sequence converges

Let $u_n$ be a bounded sequence of real numbers. Suppose that $$\lim_{n \to \infty} u_n + \frac{u_{2n}}{2} = 1$$ Show that $u_n$ converges. Can someone provide some hints or insight to this ...
0
votes
1answer
48 views

Change of order of limit and function

Let $\Omega\subset\mathbb{R}^n$ be a open and bounded domain. Suppose that $f(x)$ is a $C^1$ function for $x\in\Omega$ and $\{ x_k \}_{k=1}^{k=\infty}\in\Omega$ is a sequence with ...
0
votes
2answers
57 views

Equality of two expressions

Is the following true for even integer n = 2m > 1? $\sum_{k=1}^m 2^{2k-1} \left( {\begin{array}{*{20}c} n \\ 2k-1 \\ \end{array}} \right) = \sum_{k=1}^m 2^{2k} \left( {\begin{array}{*{20}c} n ...
2
votes
1answer
65 views

Algebraic mean problem

The Question is: $27pqr \geq (p+q+r)^3$ and $3p+4q+5r=12$, then what is the greatest value of $p^3+q^4+r^5$? How do i solve this problem? Im think harmonic mean has to be used along with geometric ...
0
votes
2answers
67 views

Proof of $(1-e^{ix})^{-1}$

In G.H. Hardy's book 'Divergent Series' there is a claim that $(1-e^{ix})^{-1} = \frac {1}{2} + \frac {1}{2} i \cot (\frac {1} {2} x) $ I, for the life of me, can't get past showing that ...