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
2answers
1k views

Complex numbers and geometric series

a) Use the formula for the sum of a geometric series to show that $$\sum _{k=1}^n\:\left(z+z^2+\cdots+z^k\right)=\frac{nz}{1-z}-\frac{z^2}{\left(1-z\right)^2}\left(1-z^n\right),\:z\ne 1$$ I thought ...
0
votes
0answers
247 views

Abel's test and Leibnitz's test

Hi does anyone know how Abel's test and Leibnitz's test( also called the 'alternating series test' for convergence) are related? Is the alternating series test sometimes called Abel's test? The ...
2
votes
1answer
197 views

Logarithmic Series Evaluation

I was trying to generate a direct formula for this series but I am not sure whether it is possible to do so. $$1\ln(1) + 2\ln(2) + 3\ln(3) + 4\ln(4)+\dots+(n-1)\ln(n-1) + n\ln(n)$$
5
votes
1answer
98 views

Convergence of the sequence $\alpha_n = \frac{1}{\sqrt{n^2 + 1}} + \dots+\frac{1}{\sqrt{n^2+n}}$

How to determine the convergence of this sequence? $$\alpha_n = \frac{1}{\sqrt{n^2 + 1}} + \frac{1}{\sqrt{n^2+2}}+ \dots +\frac{1}{\sqrt{n^2+n}}$$ I was trying to show first that the sequence has a ...
13
votes
3answers
269 views

Computing $\lim_{n\to\infty} \frac{\sqrt{n}}{4^{n}}\sum_{k=1}^{n} \binom{2n-1}{n-k}\frac{ 1}{(2k-1)^2+\pi^2}$

What tools would you recommend me for computing the limit below? $$\lim_{n\to\infty} \frac{\sqrt{n}}{4^{n}}\sum_{k=1}^{n}\frac{\displaystyle \binom{2n-1}{n-k}}{(2k-1)^2+\pi^2}$$ As soon as any ...
0
votes
1answer
20 views

Find a series $f(r)=\sum_{0}^{\infty}a_nr^n$ s.t converges for $|r| < R$ and s.t. $\lim_{r\rightarrow R-} f(r)$ exists but series does not converge.

Find a series $f(r)=\sum_{0}^{\infty}a_nr^n$ s.t converges for $|r| < R$ and s.t. $\lim_{r\rightarrow R-} f(r)$ exists but series does not converge. How do I approach this? I think that $a_n r^n$ ...
2
votes
1answer
954 views

Clarification: Proof of the quotient rule for sequences

My Problem I am currently looking for a proof for the quotient rule for sequences: $a_n$ and $b_n$ are two sequences with the limes a,b. So: When $ a_n \rightarrow a$ and $ b_n \rightarrow b$ ...
-1
votes
3answers
55 views

Test of (absolute) convergence

I have to test $$ \sum_{k=1}^{\infty}{ \frac {k^m}{m^k} } $$ for convergence and absolute convergence. m is an integer except 0. My idea is to use the root test, so: $$ \lim\limits_{k \to \infty} \...
3
votes
3answers
92 views

Show sequance is monotonic

Let $x>0$ (fixed) and $n$ be natural. Show that $$\displaystyle (x^n+x^{n-1}+...+1)^{\frac{1}{n}}$$ is monotonic. I tried by induction but didn't work but intuition tells me it's decreasing.
1
vote
2answers
233 views

Method for solving radius of convergence problem

Hi I am interested if the following method for solving for the radius of convergence for power series problem is a valid method: Find the radius of convergence of the following: $$\sum_{n=1}^{\infty}(...
1
vote
1answer
46 views

Show that $-2, -1, 0, i$ lies in the Mandelbrot set but that $1$ lies outside of it

The Question Let $c$ be a complex number. The complex numbers $z_n(c)$ are defined recursively by $z_1(c)=c$, $z_{n++1}(c)=(z_n(c))^2+c$ for $n\geq1$ The Mandelbrot set is defined by $M=\{c\in\...
6
votes
2answers
123 views

Characterize the type of sequence that satisfies $\prod (1-a_i) \leq c$

Consider a product $\prod_{i=1}^{n} (1-a_i)$ where $n\leq \infty$ and $a_i\in [0,1)$ for all $i$. I'm hoping to see if there exist conditions on the sequence $\{a_i\}$ so that $$\prod_{i=1}^{n} (1-...
1
vote
1answer
597 views

How to find a function to which given power series converges

How to find a function to which given power series converges Given Q :$$\sum_{n=1}^{\infty} (-1)^{n+1} n x^{2n-2} $$ by writing some terms I get $1−2x^2+3x^4−x^4$ if I take $x^2=y$ then I get $1−2y+...
4
votes
1answer
101 views

Find this sum $S$ using Real-analysis methods only

$$S = \sum_{k=1}^{\infty}\frac{2H_k}{(k+1)(k+2)^3}$$ I have tried a lot and failed, any help is appreciated. $H_k$ is the harmonic number. Thanks (real method only please)
1
vote
1answer
112 views

Lim inf and lim sup

I am looking for $$\lim\limits_{n \to \infty} \inf \left(-1 +\frac{1}{n^2}\right)^n$$ and $$\lim\limits_{n \to \infty} \sup \left(-1 + \frac{1}{n^2}\right)^n$$ My idea is to factorise $(-1)^n$ and ...
1
vote
0answers
45 views

Multiplying two sums?

(Real-analysis only) I will admit, I have posted a question similar to this, but this question's aim is to ask how to multiply the sum and integrate it. $\displaystyle \log^2(x) = (2)\sum_{k=1}^{\...
0
votes
2answers
68 views

Method of finding radius of convergence

Hi is it acceptable to evaluate the radius of convergence $R$ of this power series $$\sum_{n=1}^{\infty}(-1)^{n}n^{-\frac{2}{3}}x^{n}$$ by instead of taking $a_{n} := (-1)^{n}n^{-\frac{2}{3}}$ we take ...
6
votes
2answers
222 views

Closed form for a zeta series :$\sum^\infty_{k=2}\frac{(-1)^{k-1}\zeta(k)}{(k+2)2^{k+2}}$

It is not that diffcult to derive \begin{align} \sum^\infty_{k=2}\frac{(-1)^{k-1}\zeta(k)}{k2^k}=&-\frac{\gamma}{2}+\ln\left(\frac{2}{\sqrt{\pi}}\right)\tag{1}\\ \sum^\infty_{k=2}\frac{(-1)^{k-1}\...
2
votes
0answers
70 views

Evaluate $\int_{0}^{1} \log^2(x)\log^2(1-x)\,dx$ using non-elementary methods. [duplicate]

The task is to evaluate, using the specific given hint, and only real-analysis methods, no complex analysis is allowed in the problem. $$\int_0^1 \log^2(x)\log^2(1-x)\, dx$$ The hint given is: $$ \...
5
votes
1answer
144 views

How do I find the radius of convergence of a power series like this

Given: $$ a_n= \begin{cases} \frac{1}{3^n} & \text{if $n$ is prime,}\\ \frac{1}{4^n} & \text{if $n$ is not prime}. \end{cases} $$ The ratio test will work fine here, but the way series is ...
1
vote
2answers
179 views

Is there an operator for adding the numerator and denominator of a fraction separately?

Numbers in the Farey sequence are expressed as fractions e.g $F_5$: $0\over1$ $1\over5$ $1\over4$ $1\over3$ $2\over5$ $1\over2$ $3\over5$ $2\over3$ $3\over4$ $4\over5$ $1\over1$ All of the $n\over5$ ...
3
votes
1answer
102 views

Proof $(1+1/n)^n$ is an increasing sequence

I need help proving $a_n=\left(\dfrac{n+1}{n}\right)^n$ is increasing sequence on the positive integers. An exercise in the analysis book by Mattuck asks to prove $a_n=\left(\dfrac{2^n+1}{2^n}\right)^...
1
vote
1answer
88 views

equivalent metrics and uniform equivalent metrics

Let (X,d) be the Euclidean metric on the real number, and define σ(x,y)=min{1,d(x,y)} if if x, y are rationals or x, y are irrationals, and σ(x,y)=1 otherwise. I would like to study if these metric ...
6
votes
0answers
66 views

Proving another digammabinomial series result

This series is related to some extent to the previous question of mine, that is Computing $\sum_{n=1}^{\infty} \frac{\psi\left(\frac{n+1}{2}\right)}{ \binom{2n}{n}}$, where an approach by series only ...
2
votes
1answer
72 views

Formula for partial sum of (10)/(10n+1)

I'm trying to find $S_n$ of an infinite series, and I'm having trouble. Here is the equation: $$\sum_{n=1}^\infty \frac{10}{10n+1}$$ This gives me these terms: $$S_n = \frac{10}{11}+\frac{10}{21}+\...
0
votes
1answer
100 views

Find a formula for the sum of the $n$ terms of the sequence (Summations & Sequences) [duplicate]

Find a formula for the sum of the $n$ terms of the sequence: $1, 1 + 2, 1 + 2 + 2^{2}, 1 + 2 + 2^{2} + 2^{3}, ...$ My Approach: When $n$ increases the sequence increases by $2^{n}$ for every $n$. ...
1
vote
2answers
67 views

What can we say If lim U(n+1) - U(n) = 0?

So i proved that $$\lim_{n\to \infty} U(n+1) - U(n) = 0$$ and we also have $a< U(n) < b$ What can we say? Does this prove that the serie converge?
1
vote
1answer
36 views

How to solve this using integral test

Sum is from 2 to infinity and nth term is given by $ \log({ \frac{n+1}{n-1}} )\times n^{-1/2} $ Any ideas or hints will be appreciated.
1
vote
1answer
36 views

How to show this series is conditionally convergent $(-1)^{n+1}\log(n+1)$

How would one show that the infinite series $$\sum_{n=1}^\infty\frac{(-1)^{n+1}}{\log(n+1)}$$ is conditionally convergent? Tried ratio,root,nth ,logarithmic,condensation, but all in vain
0
votes
1answer
168 views

How to find Radius of convergence for power series whose coefficients are defined recursively

P1 : The Radius of convergence of power series whose nth term is given by $a_n{z^{n^2}}$ where $a_0 = 1 $ , $ a_n = 3^{-n} (a_{n-1})$ P2 :Given $a_n ={1/3^n} $, if n is prime $a_n = 1/...
0
votes
2answers
51 views

Show the convergence of a series. Telescoping?

Given the series: $\sum _{k=1}^{\infty }\:\frac{k}{k+200}$ Decide if: it is convergent or divergent
0
votes
1answer
50 views

How to find Radius of convergence for this power series

Given nth term of power series is $4^n{x^{n^2}}$ tried ratio and root test but coudnt get to solution
1
vote
2answers
35 views

Is there a sequence that each positive real number is a partial limit for it?

At the end of my last lecture in Calculus about Subsequences, the lecturer asked us whether there is a sequence that each positive real number is a partial limit for it, and no one could give an ...
0
votes
2answers
60 views

What's the formula for this sequence?

What is the generic formula for this calculation? For example, if n = 3 and x = 12000, the expanded formula looks like this: 12000 * 1 + 12000 * 2 + 12000 * 3 = 72000 n increases by 1 and is ...
0
votes
1answer
35 views

Series Convergence

Suppose $|a_n| \leq b_n - b_{n+1}$ where $b_n$ decreases monotonically to zero. Prove that $\sum_{n=1}^{\infty} a_n$ converges absolutely. My thoughts were $\sum_{n=1}^{\infty} |a_n| \leq \sum_{n=1}^{...
0
votes
3answers
46 views

Showing that an infinite series converges at a certain values of n

How could I go about showing that: $$1.4\le\sum_{i=1}^n 1/i^2\le2$$ for all $n\ge4$? I feel like turning the sum into an integral by taking a limit of the sum would be helpful but I'm not sure how to ...
-1
votes
1answer
42 views

If a sequence is in an open interval, is its limit in that interval?

If we have $(U_n)$ converges and its in some interval, can we say that $l = \lim U_n$ is in that interval? I know that in this case that $\lim U_n = \sup \{U_n\}$ but it appears that the supremum of $...
0
votes
0answers
65 views

Power Series of the Principal Branch of Logarithm

How would I go about determining the power series expansion for the principal branch of $\log$ about $z = i$? I would assume I should start from $\frac{1}{z}$, but I don't know how to manipulate this ...
1
vote
1answer
31 views

show series converges by root test

Let $x$ be a real number with $|x|< 1$, and $q$ be a real number. Show that the series $\sum\limits_{n=1}^\infty n^qx^n$ is absolutely convergent, and that $\lim\limits_{n\to \infty} n^qx^n= 0$ My ...
1
vote
0answers
60 views

How to find $\sum_{k=1}^n k^k$?

Actually question which I found: Find the sum of the series $1^1+2^2+3^3+ \cdots +n^n $ This question has been bothering me since a long time. Any help would be appreciated!
7
votes
3answers
143 views

Proving bounds on an infinite product

Let $p$ be an infinite product, such that $p = 2^{1/4}3^{1/9}4^{1/16}5^{1/25} ...$ Prove that $2.488472296 ≤ p ≤ 2.633367180$. I start this problem by representing p in the infinite product ...
3
votes
1answer
115 views

Divergent Sequence for Wau

So I just "learned" about the number Wau from Vi Hart's video. It's amusing, to be sure, but the actual "definition" she presents got me thinking. We can formalize the construction in this way: set $...
2
votes
2answers
48 views

Show that sequence has limit

We know that(1) $\displaystyle \lim_{n \to \infty}{(a_{n+1}-a_n)}=0$ and (2) $\displaystyle |a_{3m}-a_{3n}|<\varepsilon$ show that $a_n$ converge and explain why it's not sufficient to converge ...
1
vote
2answers
46 views

If the ratio of AM and HM of two positive real numbers (a and b) is m:n. Find a:b.

$$AM = \frac{a+b}2\text{ and }HM = \frac{2ab}{a+b}$$ Therefore $AM/HM= (a+b)^2/4ab=m/n$. I'm not able to proceed further. What do I need to do to get $a/b$? Thanks in advance. :) Also, the answer ...
2
votes
1answer
165 views

The limit of products of the form $(n^3-1)/(n^3+1)$

Calculate $$\lim_{n \to \infty} \frac{2^3-1}{2^3+1}\times \frac{3^3-1}{3^3+1}\times \cdots \times\frac{n^3-1}{n^3+1}$$ No idea how to even start.
7
votes
2answers
151 views

Proof $1 -\frac{1}{5} + \frac{1}{9} - \frac{1}{13} + … = \frac{\pi + 2\ln(1+\sqrt2)}{4\sqrt2}$

I'm trying to show that $$1 -\frac{1}{5} + \frac{1}{9} - \frac{1}{13} + \cdots = \frac{\pi + 2\ln(1+\sqrt2)}{4\sqrt2}.$$ I thought of using the power series for $\tanh^{-1}z$ which I found was $\...
7
votes
1answer
138 views

Find the closed form of the digamma related series

The question I asked here Computing $\sum_{n=1}^{\infty} \left(\psi^{(0)}\left(\frac{1+n}{2}\right)-\psi^{(0)}\left(\frac{n}{2}\right)-\frac{1}{n}\right)$ made me think to ask for your support for ...
0
votes
1answer
55 views

Relation between sum and integral

I have an exercise (from physics) where I am supposed to show $$\sum_{k'<k_f} \frac{1}{|k-k'|^2} = C \left( \frac{1}{2} + \frac{1-(\frac{k}{k_f})^2}{4 \left( \frac{k}{k_f} \right) } ln |\frac{1 + ...
0
votes
1answer
64 views

Limit of product of elements of sequence

Given sequence $a_n = \sqrt[n]{(\frac{2012}{2013})^n - (\frac{2011}{2012})^n}$ and $A_n$ such $A_1 = 17$, $A_{n+1} = A_n \cdot a_n$. I have to examine limit of $A_n$. $\lim_{n\to\infty} a_n= \frac{...
1
vote
2answers
83 views

What is $\lim_{n\rightarrow\infty}\frac{n^{\sqrt{n}}}{2^{n}}=? $ [duplicate]

I'm trying to calculate the following limit: $$\lim_{n\rightarrow\infty}\frac{n^{\sqrt{n}}}{2^{n}}$$ I assume it's equal to $0$, but can't find a way to prove it. I don't even know where to start..