13
votes
2answers
492 views

Intuitive ways to get formula of cubic sum

Is there an intuitive way to get cubic sum? From this post: combination of quadratic and cubic series and Wikipedia: Faulhaber formula, I get $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$ I think ...
1
vote
1answer
24 views

find a $B_{n,j}$ such that $|A_{n,j}-L_j| \leq B_{n,j}$ $\forall n,j$ and $\sum_{j=0}^{\infty}B_{n,j}$ converges

We have $A_{n,j}= 3(-1)^j2^{n-j+1}\frac{(2(n-j)-4)!}{(n-j)!(n-j-2)!}\binom{j+2}{2}\frac{n^\frac{5}{2}}{8^n}$ and $L_j=(-\frac{1}{8})^j\binom{j+2}{2}\frac{3}{8\sqrt{\pi}}$ So I know $\lim_{n \to ...
7
votes
3answers
164 views

Calculate $\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$

I'm an eight-grader and I need help to answer this math problem. Problem: Calculate $$\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$$ This one is very hard for me. It ...
1
vote
1answer
70 views

combination of quadratic and cubic series

I'm an eight-grader and I need help to answer this math problem (homework). Problem: Calculate $$\frac{1^2+2^2+3^2+4^2+...+1000^2}{1^3+2^3+3^3+4^3+...+1000^3}$$ Attempt: I know how to calculate ...
2
votes
1answer
146 views
+50

An arithmetic sequence whose members do not contain the digit ‘9’

There is a non-constant arithmetic progression made of natural numbers only; none of them contains the digit $9$. Prove that such an arithmetic progression has no more than $72$ terms.
2
votes
3answers
57 views

Find the 325th term of the series 7,16,25,34…

One of my friend gave me the series 7,16,25,34,43... I figured it out easily that the sum of digits is 7 in each case. How can I find the 325th term of this series? Also is there any trick/formula to ...
4
votes
2answers
59 views

what is the sum of this series: $\frac{2}{\pi}\Sigma_{k=1}^{\infty}\frac{(-1)^{k-1}}{2k-1}$

Can anyone help me with this? What is the sum of this series: $\frac{2}{\pi}\Sigma_{k=1}^{\infty}\frac{(-1)^{k-1}}{2k-1}$ I got it after plugging $x=-1$ in a fourier series Thank you!
0
votes
3answers
278 views

What is the probability that A will win…

Two players are rolling two dices, if they get 6 Player A wins, if they get 7, player B wins, else they rolling the two dices again... What is the probability that A will win? I'd like to get any ...
1
vote
1answer
72 views

How to solve this graphing question?

$ \frac{|x-2|} {(x^2-4)}+\frac{(x-2)} {|x-2|} = b $ determine for which values of $b$ the equation has one and only solution. I tried sketching the graph, but was unable to do so accuratly...also, ...
4
votes
2answers
148 views

Combination of quadratic and arithmetic series

Problem: Calculate $\dfrac{1^2+2^2+3^2+4^2+\cdots+23333330^2}{1+2+3+4+\cdots+23333330}$. Attempt: I know the denominator is arithmetic series and equals ...
2
votes
4answers
457 views

Proving 7n+5 is never a cubic number?

This is from a question that starts with: An arithmetic progression of integers an is one in which $a_n=a_0+nd$, where $a_0$ and $d$ are integers and n takes successive values $0, 1, 2, \cdots$ Prove ...
4
votes
5answers
226 views

How to find $\sum_{k \in \mathbb{Z}}\frac1{(k+a)(k+b)}$

Let $a,b$ be two unequal integers. I have to find the sum below. $$ \sum_{k \in \mathbb{Z}}\frac1{(k+a)(k+b)} $$ I should use complex analysis, but I have no clue where to start. I only now that I can ...
12
votes
3answers
206 views

How to show that $ \sum_{n = 0}^{\infty} \dfrac {1}{n!} = e $

How to show that $$ \sum_{n = 0}^{\infty} \dfrac {1}{n!} = e $$ where $e = \lim \left({1 + \dfrac 1 n}\right)^n$ I'm guessing this can be done using the Squeeze Theorem by applying the AM-GM ...
1
vote
1answer
46 views

Need help considering series like these: $\sum_{n=1}^\infty\langle x,e_n\rangle e_n$

I'm working in a Hilbert space $H$ with ONB $(e_n)$ and I have $\alpha=(\alpha_n)\in\ell^\infty$. I have an operator that looks like this: $$T_\alpha x=\sum_{n=1}^{\infty}\alpha_n\langle ...
6
votes
2answers
525 views

Sum of this series

$$ \mbox{How do I find the sum of this series}\quad \sum_{n=0}^{\infty}{\sin^{3}\left(3^{n}\right) \over 3^{n}}\ {\large ?} $$ Hints in the right direction would be appreciated.
0
votes
2answers
37 views

It is a question about uniform convergence of a function.

I solved this problem . Is my answer a correct ? $$ x \in [0,\infty) ,\lim_{n \to \infty} \frac{nx}{1+n^2x^2}=0\ \ \ $$ Is $ \frac{nx}{1+n^2x^2} $ converged uniformly on $0$ ? My solution $$ ...
9
votes
2answers
107 views

Divergent of a vector field on a sequence of spheres

I'm studying for my exams and I found this problem in the book "Advanced Calculus", written by Friedman: "Consider a sequence of spheres $S_n$ in $\mathbb{R}^3$ with center $P_n$ and radius $r_n$, ...
1
vote
2answers
76 views

Taylor series of the function $f(x) = (1+x) ^{\frac{1}{x}}$

Good night!! I got this problem and I'd like to find all the mistakes in my statement. This is my prblem. Find the binomial coefficients of $f(x) = (1+x)^{\alpha}$, with $\alpha \in \mathbb{R}$, and ...
3
votes
2answers
84 views

Are there any 3 natural numbers that satisfy $a^2+b^2=2z^2 $?

Are there any 3 natural numbers that satisfy $a^2+b^2=2z^2 $? This is a question that arised as I was trying to solve another question: Is there an arithmetic progression, of natural numbers in which ...
2
votes
0answers
81 views

Questions about central polygonal numbers $1, 2, 4, 7, 11, 16, 22, 29, 37, 46,\cdots$

Formula for Central polygonal numbers is $\frac{n(n+1)}{2} + 1$, if $n=1$ or $n$ is prime, we get the new sequence $A$: 2, 4, 7, 16, 29, 67, 92, 154, 191, ... It seems that all primes either is ...
0
votes
2answers
40 views

$f_n:[0,1]\to [0,1]$ continuous, $f_n\to f$ uniformly, prove: ${\frac1n}\sum_{k=1}^{n}{f_k} \to f$ uniformly

I was able to prove, hopefully correctly, that the sum converges uniformly. But, I'm not sure how to show it converges uniformly specifically to $f$. The way I proved it converges uniformly was by ...
9
votes
1answer
119 views

Convergence of $\sum_{n=1}^{\infty}\frac{1}{3^n\ \sin(n)}$

Does this series converge? Root test and ratio test are inconclusive.
0
votes
1answer
37 views

Determine the value of the following series.

Find the partial sum $S_n$ of the telescoping series $$\sum _{ n=1 }^{ \infty }{ \frac { 1 }{ \left( n+2 \right) n! } } $$
3
votes
1answer
112 views

Finding some counterexamples

The sequence of $\begin{Bmatrix} {x}_{n} \end{Bmatrix}$ is strictly decreasing,$\quad \lim_{n\to\infty }{x}_{n}=0 \quad $,and$\quad \lim_{n\to\infty }{y}_{n}=0 .$ From the above ...
0
votes
1answer
27 views

Mathematics - geometric progression question

If $a$, $b$ and $c$ are in geometric progression, then what are $\log_ax$, $\log_bx$ and $\log_cx$ in? What I did: I substituted values for $x, a, b$ and $c$ and tried to solve it further. What I ...
0
votes
3answers
74 views

Infinite sum of 1 over a quadratic [closed]

I want to find the following sum $$\sum_{n=1}^{\infty}\frac{1}{(4n-3)(4n-1)}$$ This was given to us as homework by our teacher while teaching telescoping series but i guess this doesn't telescope.. ...
0
votes
1answer
41 views

Given a power series

Consider the power series $\displaystyle\sum^{\infty}_{n=2} \frac{(-2)^n}{n(n-1)}x^n$ Let $ f(x) $ describe the sum function of the series on $ (-r,r) $. Argue that $f$ is two times differentiable ...
0
votes
1answer
37 views

Taylor Series expansion and first four terms of $7x^2 e^{-4x}$

As the series I got $$ \sum_{n=0}^\infty (-1)^n(4x)^n/n! $$ which I think is right. However, I am not sure how to get the first four non zero terms.
6
votes
3answers
147 views

Consider convergence of series: $\sum_{n=1}^{\infty}\sin\left[\pi\left(2+\sqrt{3}\right)^n\right]$

Consider convergence of series: $$\sum_{n=1}^{\infty}\sin\left[\pi\left(2+\sqrt{3}\right)^n\right]$$ My tried: We have $$\sum_{n=1}^{\infty }\sin(\pi ...
2
votes
1answer
18 views

an AP is changed to form a GP

Three numbers whose sum is $15$ are successive terms of an arithmetic series. If $1, 1$ and $4$ are added to these three numbers respectively, the resulting numbers are successive terms of a geometric ...
4
votes
2answers
325 views

The product of two divergent series is divergent?

This is an TRUE/FALSE queston: The product of two divergent series is divergent. The correct answer is FALSE. I know that the product of two convergent series may not be convergent (i.e. ...
2
votes
2answers
42 views

$\sum_{n=1}^{\infty}(a_{2n-1}+a_{2n})$ is convergent and $\lim_{n\to\infty}a_n=0$ What about $\sum_{n=1}^{\infty}a_n$?

Suppose that $$\sum_{n=1}^{\infty}(a_{2n-1}+a_{2n})$$ is convergent, and $$\lim_{n\to\infty}a_n=0.$$ Prove that $\sum_{n=1}^{\infty}a_n$ is convergent. I know if a series is convergent then ...
1
vote
1answer
33 views

Show that $\sum_{k=1}^N\frac{1}{(k+a)(k+b)}=\frac{1}{b-a}\sum_{a<k\leq b}\frac{1}{k}-\frac{1}{b-a}\sum_{a<k\leq b}\frac{1}{k+N}$

I am quite stuck on this problem and I don't know how to proceed. The question states: Let $a,b,N\in\mathbb{N}$, $b>a$, $N\geq b-a$. Show that ...
2
votes
2answers
84 views

Prove $\sum_{n=1}^{\infty}\frac{1}{2^n+1}$ convergence

Prove that $$\sum_{n=1}^{\infty}\frac{1}{2^n+1}$$ convergence by Cauchy theorem: $$\forall\varepsilon>0, \exists N>0: \text{whenever}\ n>N,\ |a_{n+1}+\cdots+a_{n+p}|<\varepsilon,\ p=1, ...
0
votes
1answer
36 views

Limit of a sequence defined by a recurrence relation

I have to find the limit of the sequence defined as follows: $u_0>0$ $u_{n+1} = \sqrt{u_n+\sqrt{u_{n-1} + \ldots + \sqrt{u_0}}}$ I really have no idea for that one... Not even how to start. ...
3
votes
2answers
90 views

Convergence of infinite series involving $\frac{\sin(x)}x$. [closed]

Show that the infinite series $$\sum_{x = 1}^{\infty} \frac{\sin(x)}{x}$$ is convergent. Please answer so that a Calculus student can understand.
0
votes
2answers
32 views

Pointwise and uniform convergence of this series

$$\sum_{n=1}^{\infty}\left(1- \frac{1}{2n}\right)^{-n^2}(x^2-1)^n$$ I've tried treating it as a power series centered around $x = 1$ and $x = -1$ and using root test I arrive to radius of convergence ...
0
votes
1answer
40 views

How do you formally prove $\limsup\limits_{n\rightarrow\infty}(|na_n|^{\frac{1}{n}}) = \limsup\limits_{n\rightarrow\infty}(|a_n|^{\frac{1}{n}})$

My definition of $\limsup$ is it is the supremum of the accumulation points of a sequence (i.e the supremum of the limits of all possible subsequences of a sequence). So if: ...
0
votes
0answers
22 views

Finding coefficient $c_n$ of Cauchy product for $a_k = (\frac{-1}{2})^k, b_k = \frac{(-1)^k}{k+1}$

I need to find the coefficient $c_n=\sum_{k=0}^n a_k b_{n-k}$ for the Cauchy product $\sum_{k=0}^\infty c_k=(\sum_{k=0}^\infty a_k)(\sum_{k=0}^n b_k)$ with $(a_k)=(\frac{-1}{2})^k$ and $(b_k) = ...
4
votes
2answers
29 views

formula for number triangles

Hi, I have a triangle starting from $0$ and going up by one on the bottom row until there are $r$ items on the bottom row and there are $r$ rows a number is formed by adding the two numbers towards ...
1
vote
2answers
28 views

Upper bound for geometric type equation

I have a problem with my homework and don't see the answer. Does anyone know of conditions on $a$ ensuring the convergence of $$\sum\limits_{n=0}^{\infty} \sum\limits_{k=0}^{n-1} a^k?$$ Using the ...
3
votes
4answers
94 views

Sum the following $\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)} $

Evaluate: $$\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)} $$ I rewrote the sum as $$\sum_{n=0}^{\infty} \frac {1}{4^{8n-7}(8n-7)} - \sum_{n=0}^{\infty} \frac {1}{4^{8n-3}(8n-3)}$$ Now, I ...
1
vote
2answers
53 views

How to determinate ending point using series?

Consider that I have a path which start at point (0,0). I need to find the ending point of the path using series. The path look like this : Any idea on how to start that ?
-1
votes
2answers
25 views

Find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$

Let $\alpha$ and $\beta$ be two given constants, how to find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$, where $c_0 = {\alpha}^{\beta}$.
2
votes
5answers
125 views

Find a closed form for $\sum_{k=0}^{n} k^3$ [duplicate]

Find a closed form for $\sum_{k=0}^{n} k^3$. I would appreciate ideas for approaching questions like this in general as well. Thanks.
0
votes
0answers
102 views

Find all the subsequences which are arithmetic progression

Given a sequence is there a linear or sub-linear algorithm to find all the sub-sequences that are arithmetic progressions with a given D, where D is the consecutive difference between the elements ?
3
votes
3answers
94 views

Prove $a_n=1+\frac{a_{n-1}}{1+a_{n-1}}$ increasing

There is a homework question in Calculus-1 course: Calculate the limit of $\{a_n\}$: $$a_1=1,\ a_n=1+\frac{a_{n-1}}{1+a_{n-1}}$$ I think the key points are bounded and increasing, and I have proved ...
3
votes
3answers
117 views

Is this question of sequence a Mathematical one, i.e. does it have objectively only one answer for each subpart.

This question is taken from 11th class Math book. Look at this question: At the very first glance one can tell that all the three sequences are G.P But! by using interpolation(as this answer ...
0
votes
2answers
34 views

Integral Test for Convergence - Log of a Log

I need to investigate the convergence of the series $\sum_{n=3}^{\infty}\dfrac{1}{n\ln n}$ So, doing the integral test, I end up with (shortcutted because integration is boring): ...
11
votes
4answers
1k views

How to solve this sequence $165,195,255,285,345,x$

This is a question appeared in a competitive exam. The question is: Find the unknown term in $165,195,255,285,345,x$ 1)375 $\ \ \ \ \ \ \ \ $ 2)420 3)435 $\ \ \ \ \ \ \ ...