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

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0
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1answer
30 views

Prove that $\sum_{x=0}^{\lambda - 1}(\frac{\lambda^{x+1}}{x!}-\frac{\lambda ^x}{(x-1)!})=\frac{\lambda^\lambda}{(\lambda - 1)!}$

Prove that $\displaystyle\sum_{x=0}^{\lambda - 1}\left(\frac{\lambda^{x+1}}{x!}-\frac{\lambda ^x}{(x-1)!}\right)=\frac{\lambda^\lambda}{(\lambda - 1)!}$. I think I need to write it in the derivative ...
0
votes
1answer
76 views

Calculate $\sum_{n=1}^{\infty}n^2q^{n-1}$

Please show me how to calculate the sum of this infinite series: $$\sum_{n=1}^{\infty}n^2 q^{n-1}$$ I should have included the condition $\mid q\mid$<1 And I was able to solve the infinite ...
3
votes
1answer
52 views

Conjecture about a property of concave functions

Trying to prove a proposition in my paper, which can potentially use a conjecture about convex (concave) functions. This is likely to be wrong. I appreciate any thoughts on how to prove/disprove this. ...
5
votes
3answers
235 views

Can rearranging a SEQUENCE (not a series) change the limit?

I have this question on a homework assignment. I sat down with two other people for a long time and we derived the alternating harmonic series example, but I don't think that's valid because the ...
0
votes
2answers
113 views

Nonexistence of Limit of Sum of Prime Factors

In trying to prove the following problem, I find great difficulty in proceeding to generalizing some results: Let $s(n)$ be the sum of prime factors of an integer $n$. Prove that $\lim_{n \to \infty} ...
2
votes
1answer
241 views

approximation of sum of gaussian-like function?

Let: $g(u; x,s) = \dfrac{1}{s\sqrt{2\pi}} \exp\left(-\dfrac{1}{2} \left(\dfrac{x-u}{s}\right)^2\right)$ Where $x,s$ are parameters I'm looking for a closed-form solution or approximation of: ...
0
votes
4answers
124 views

Sum of series proof

How can I find the sum of $\sum_{i=1}^{n} i2^i$ in a closed form. I have a feeling I can do this using differentiation of an existing series but cant seem to find it.
1
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1answer
768 views

how to plot sequence of partial sums in Maple

I've seen other questions and tried to use internet and the built-in help but I'm still at a loss. Given the sequence seq((n,1/2^n),n..20)), I want to plot the corresponding sequence of partial sums, ...
8
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3answers
211 views

compute $1-\frac 12+ \frac15 - \frac 16+ \frac 19- \frac{1}{10}+ \cdots + \frac{1}{4n+1}-\frac{1}{4n+2}$

Knowing that $1 - \frac 12 + \frac 13 - \cdots = \ln 2$ and $1 - \frac 13 + \frac 15 - \cdots = \frac{\pi}{4}$, compute $1-\frac 12+ \frac15 - \frac 16+ \frac 19- \frac{1}{10}+ \cdots + ...
9
votes
2answers
366 views

What's the limit of $(1-1/2)(1-1/4)(1-1/8)…$?

I know that $$\prod_1^\infty \left(1-\frac{1}{2^n}\right)$$ converges to a positive number because the series $\sum 2^{-n}$ is convergent. Do we know the limit? If so, how? Aside: I am ...
2
votes
3answers
734 views

Monotone and Bounded Sequences: Proof

Say $s_1=1$ and $s_{n+1}=\frac{1}{5}(s_n+7)$ for $n\geq 1$. Prove that the sequence is monotone and bounded, then find the limit.
4
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1answer
103 views

What is the name for defining a new function by taking each k'th term of a power series?

With the definitions of the three functions $$ f(x)= 1 + \frac{x^3}{3!} + \frac{x^6}{6!} + ... \\ g(x)= x + \frac{x^4}{4!} + \frac{x^7}{7!} + ... \\ h(x)= \frac{x^2}{2!} + \frac{x^5}{5!} + ...
0
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2answers
68 views

Binomial and series with 2 coefficients

I would be very grateful if you would help me with this question: Find the sum : $$ \sum_{k=0}^{n}\binom{2n}{k} $$
1
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0answers
41 views

on the interval $[-1,1]$, $\sum (-1)^n {x^2+n^2\over n^3}$ is absolutely convergent?

on the interval $[-1,1]$, $\sum (-1)^n {x^2+n^2\over n^3}$ is absolutely convergent right? $f_n(x)={x^2+n^2\over n^3}$ is uniformly convergent by Dinis Theorem as they are monotone and continuos on ...
0
votes
3answers
93 views

Series convergence

So I have a test next week and I tried to solve this question but I do not know how. Regarding $ \sum_{n=1}^{\infty} \frac{xn}{e^{nx}}$ Proof that it is Uniform convergence in $[\delta , \infty)\ \ ...
1
vote
1answer
89 views

A question on Cauchy sub-sequences in a metric space $(X,d)$

Let $(X,d)$ be a metric space, and let $(x_n)$ be a sequence in $X$. Prove that if $(x_n)$ has a Cauchy subsequence, then for any decreasing sequence of positive $\epsilon_k \rightarrow 0$, there is a ...
4
votes
3answers
263 views

Discuss the convergence of $ \left \{ a_n \right\} $ where $ a_{n+1}=\frac{a_0}{2}+\frac{a_n^2}{2},n\geq 1 $

Let $$ a_{n+1} = \dfrac{a_0}{2} + \dfrac{a_n^2}{2} $$ where $ a_1 = \dfrac{a_0}{2} $ and $ n\geq 1 $ Discuss the convergence of $ \left\{a_n\right\} $
2
votes
3answers
248 views

If Q-Cauchy sequences must limit to a rational, how can they construct the reals?

I'm currently in a graduate Math for Economists course, and we spent last week learning how to construct the reals from the rationals using Cauchy sequences and their equivalence classes. I know that ...
0
votes
1answer
107 views

Sum of number of factors of first N numbers [duplicate]

Given a number N ( Value can be large like N < 10^9 ) How can we calculate sum of the number of factors of first N numbers?? Example : For n = 3 Answer: = #f(1) + #f(2) + #f(3) --- { #f(n) ...
2
votes
1answer
74 views

Show in between steps in this Riemann zeta function equivalence/reduciton

In the answer chosen by the OP in this question I had trouble understanding the steps taken to get the equivalences/reduce the zeta function into another one. Can somebody show me the steps to go from ...
-2
votes
1answer
116 views

Study the following sequences and determine if they are bounded

Study the following sequences and determine if they are bounded: 1) $a_n = \frac{4}{n+3}$ 2) $a_n = \frac{2n}{n+1}$ 3) $a_n = \frac{n^2 + 2}{2n^2+1}$ 4) $a_n = 2n + 3$ 5) $a_n = \frac{ ...
0
votes
2answers
363 views

Absolute Convergence $ \sum_{n\geq1} \frac{(-1)^n}{\sqrt[3]{n^2+2}}$

Like always. New example, and I have no idea. Calculating with Series for 2 weeks now, seing now progress. Its fun. I need to check for absolute convergence and convergence of this series: $$ ...
1
vote
3answers
659 views

Prove that a sequence is a Cauchy sequence.

The question is this. Let $(s_n)$ be a sequence such that $$\left|s_{n+1}-s_n\right| < 2^{-n}, \forall n \in N$$ Prove that $(s_n)$ is a Cauchy sequence and hence a convergent sequence. My proof ...
3
votes
2answers
147 views

Prove that if $a_{n+1} = a_n^2$, the last $n$ digits of $a_{n+1}$ are the same as the last $n$ digits of $a_n$.

I have been working on this problem for a while. I know that I have to prove it using induction, but I'm unsure of the next step. The formula for the terms is: $a_{n+1} = 5^{2n}$ with $a_1 = 5$. The ...
-1
votes
1answer
186 views

Predict the next four numbers in a series of numbers.

List the next four numbers in this sequence: 1,4,9,1,6,2,5,3,6,4,9,6,4,8,1 What is the pattern?
0
votes
1answer
85 views

Behavior of sequence: $a_n=ka_{n-1}-n$

Just a simple problem that has gotten me curious. For fixed $k > 1$, consider the sequences defined by: $a_n=ka_{n-1}-n$ For high values of $a_0$, the sequence diverges to $+\infty$. For low ...
0
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6answers
1k views

Infinite amount of additions, finite sum?

I suggest it's a popular question, so if it was asked already (I couldn't find it anyway), close this question instead of downvoting, thanks! Let's check this addition: ...
0
votes
1answer
63 views

What about the convergence of these series?

$$ \sum_{n=1}^{\infty} \arctan \frac{1}{2n + 1} $$ $$ \sum_{n=1}^{\infty} (\frac{\pi}{2} - \arctan ( \log n ) ) $$ $$ \sum_{n=1}^{\infty} \sin ( n \pi + \frac{1}{\log n } ) $$
2
votes
0answers
151 views

Does this series that has terms $1/n$, then terms $1/n (\log n)^2$, then terms $1/n \log n (\log \log n)^3$, etc. converge or diverge?

Define $\log_{(k)}$ to be the logarithm function iterated $k$ times, where $\log_{(0)}$ is the identity function. Consider the series $\sum_n 1/a_n$ where $$a_n = (\log_{(f(n))} n)^{f(n)+1} ...
0
votes
2answers
30 views

Convergeny Again - Direct Comparison

It's me again. Somebody please tell me if its considered as spam what i do here for asking a question so often :) $$ \sum_{n\geq0} \frac{n+3}{7n^2-2n+1}$$ I assume that this series diverges as it ...
5
votes
2answers
290 views

Find all positive integers $n$ for which $1 + 5a_n.a_{n + 1}$ is a perfect square.

The sequence $a_1, a_2, \ldots $ is defined by the initial conditions $$a_1 = 20; \quad a_2 = 30$$ and the recursion $$a_{n+2} = 3a_{n+1} - a_n$$ and for $n \geq 1$. Find all positive integers $n$ ...
0
votes
1answer
43 views

complex sequences

my series and sequence knowledge has gone a little rusty so I was wondering if you could help me on the right path here. The assignment is to calculate the sum of the series (1/8)^n * e^(j(npi)/8) as ...
1
vote
0answers
53 views

Can this limit be proven to converge to the Logarithmic Integral?

Here's the limit: $\displaystyle\lim_{k \rightarrow 1^+}\sum_{j=0}^{\lfloor\log_k x-\log_k\mu\rfloor}\frac{k^{j+\log_k\mu}}{j+\log_k\mu}$ where $\mu=1.45136380...$, Soldners Constant. Empirical ...
1
vote
1answer
62 views

Simplify the summation

How $$\sum_{k=1}^{\infty}\frac{(-1)^{2k-2}}{(2k-1)^{2m}}+\sum_{k=1}^{\infty}\frac{(-1)^{2k-1}}{(2k)^{2m}}$$ can be rewritten as $$\sum_{k=1}^{\infty}(k)^{-2m}-2\sum_{k=1}^{\infty}(2k)^{-2m}$$ ??? ...
2
votes
2answers
236 views

Double sequence $z_{mn}$ Converges but it doesn't imply $z_{mn}$ is bounded

I have noticed an interesting thing in double sequence $z_{mn}$ and I can't see why such thing happens. Definition: Double sequence $z_{mn}$ is a mapping from $\mathbb{N}\times\mathbb{N}\rightarrow ...
0
votes
1answer
57 views

To prove the lower bound of a limit

I am thinking a limit which is very interesting: For any positive sequence $S_j$ bounded away from zero. we have the following result: $\limsup\limits_{k\to\infty}\frac{s_k}{s_l}\geq1$, where ...
1
vote
0answers
135 views

Value of sum with primes

Can anyone tell me the value of sum $\sum_p\left(\log p(\frac{1}{2p}-\psi(\frac{p+2}{2})+\psi(\frac{p+1}{2})\right)-\sum_n\frac{(\log(2^n)}{2^n}$ where $p$ ranges over prime powers and $n$ ranges from ...
2
votes
3answers
97 views

Proving that a sequence is between certain values at certain n

I'm given that $a_1=1$, and for every $n \gt1, a_{n+1} = a_n + \frac{1}{a_{n}}$. I need to prove that $20 < a_{200} < 24$. I tried finding a limit at infinity setting both limits to $L$ ( for ...
7
votes
3answers
154 views

Nice problem Which is bigger $e^a$ and $a^3$

let $$a=\sum_{n=0}^{\infty}\dfrac{\left(\dfrac{n+1}{3}\right)^n}{(n+1)!}$$ My Question: Which is bigger $e^a$ and $a^3$ I guess $$e^a=a^3$$ but I can't prove it,and I think this is ...
2
votes
2answers
2k views

Accumulation Points & Convergence: A Sequence Existence Proof

Request: Prove that $x$ is an accumulation point of a set $S$ iff there exists a sequence $(s_n)$ of points in $S\setminus \{x\}$ such that $(s_n)$ converges to $x$. Attempt: Since this is a ...
1
vote
3answers
157 views

Evaluating $\sum_{i=0}^k \frac{(-1)^{k-i}k! (n+i)^{k-1}}{i!(k-i)!}$

How to evaluate the the following sum where $n $ is an integer greater than $0$. $$\sum_{i=0}^k \frac{(-1)^{k-i}k! (n+i)^{k-1}}{i!(k-i)!}$$ I think the answer is $0$, but I can not prove it.
1
vote
1answer
108 views

Question about divergent series

Is it true that if $(a_n) \geq 0$ and $\displaystyle\sum_1^\infty a_n$ diverges, then $\displaystyle\sum_1^\infty a_n(1-r^n)$ diverges for all $r \in (0,1)$? I think it's true but I'm having a hard ...
0
votes
2answers
2k views

sum of a Fibonacci type series

Sum of $n$ (or in particular, $52$) terms of $1,6,7,13,20,33\ldots$ just cant think of any relation. do we have a direct formula for the sum of fibonacci numbers?
3
votes
1answer
94 views

If $(s_n)$ converges to $s$, then $(\lvert s_n\rvert)$ converges to $\lvert s\rvert$.

Question: If $(s_n)$ converges to $s$, then $(\lvert s_n\rvert)$ converges to $\lvert s\rvert$. Prove or give a counterexample. Attempt: The statement is true because if $(\lvert ...
1
vote
2answers
142 views

Prove $\sum_{n=1}^\infty a_n < \infty \implies \sum_{n=1}^\infty a_n(1-r^n) \to 0 \text{ as } r \to 1^-.$

Let $(a_n)$ be a sequence of positive terms and suppose $$\displaystyle\sum_{n=1}^\infty a_n < \infty.$$ Prove (from first principles) that $$\displaystyle\sum_{n=1}^\infty a_n(1-r^n) \to 0 \text{ ...
2
votes
1answer
144 views

Derive power series for $\frac{1}{1-x^2}$ from $\frac{1}{1-x}$?

The series $\sum_{k=0}^{\infty} x^k$ is known and equals $\frac{1}{1-x}$ for $|x|<1$. Can I use this fact and derive the power series for $\frac{1}{1-x^2}$ from it, eg using ...
1
vote
3answers
278 views

A sequence $(a_n)$ converges to $L$, $|L| < 1$. Prove that $(a_n)^n$ converges to $0$

So $a_n$ converges to $L$, and the absolute value of $L$ is less than $1$. How do we go about proving that the sequence $(a_n)^n$ converges to 0? I tried a couple of basic methods but I don't seem to ...
3
votes
1answer
132 views

Series of modified Bessel functions

There is a known identity to evaluate a sum of the form $$\sum_{n\geq1} \rho^n I_n(\omega) $$ Where $\rho>0$, $\omega >0$ and $I_n$ is the modified Bessel function of the first kind. ??? ...
1
vote
3answers
393 views

Convergence of recursive sequence

I have tried to do this exercise. Do you think my solution is ok? Is it possible to get more information about the convergence? Is there a better way to do it? Let $ f:[0,1]\mapsto[0,1] $ be a ...
-4
votes
1answer
215 views

The next number in the series?

This is a form of an aptitude test. I just wanted to know what comes next in this series? ...