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Question 1

As questions regarding sequences that verifies a linear recurrence relation with constant coefficients are posted very often on this site and that there appear to be no reference post about it, so I ...

Question 2

Consider a sequence $(a_n)_{n\in\mathbb N}$ defined by $k$ initial values $(a_1,\dots,a_k)$ and $$a_{n+k}=c_{k-1}a_{n+k-1}+\dots+c_0a_n$$ for all $n\in\mathbb N$. What are some ways to get closed ...

Question 3

A common "trick" for obtaining a closed form of a geometric series is to define $$ R := \sum_{k=0}^{\infty} r^k, $$ then manipulate the series as follows: \begin{align} R - rR &= \sum_{k=0}^{\...

Question 4

If $a_n$ is a sequence such that $$a_1 \leq a_2 \leq a_3 \leq \dotsb$$ and has the property that $a_{n+1}-a_n \to 0$, then can we conclude that $a_n$ is convergent? I know that without the ...

Question 5

If $\langle f_{n}\rangle$ be a sequence of positive numbers such that $$f_{n}=\frac{f_{n-1}+f_{n-2}}{2}$$ $\forall n\gt2$ ,then show that $\lt f_{n}\gt$ converges to $$\frac{f_1+2f_2}{3}$$ Replacing ...

Question 6

The recursive formula is $t_n=\frac {t_{n-1}+t_{n-2}}2$ Changing $t_1$ and $t_2$ changes the number where the sequence converges as $n \to \infty$. With the help of everyone at StackExchange, I ...

Question 7

The recursive formula is $$t_n=\frac {t_{n-1}+t_{n-2}}2$$ as $n$ approaches infinity the mean sequence converges at a certain number. Changing $t_1$ and $t_2$ changes the number where the sequence ...

Question 8

I was wondering if there is also a closed expression for the series $$\sum_{n=0}^{\infty} \binom{n+k}{k} x^n$$ where $|x|<1.$ A few examples suggest that the answer is $\frac{1}{(1-x)^{k+1}}$ ...

Question 9

How do I prove that $x_{n+2}=\frac{1}{2} \cdot (x_n + x_{n+1})$ $x_1=1$ $x_2=2$ is convergent?

Question 10

When I tried to solve this integral: $$\int_0^\infty {\frac{{{x^3}}}{{1 + {e^x}}}} \;{\rm{d}}x$$ I had trouble computing the sieries: $$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4}$$ Thanks.

Question 11

If $a_n=(a_{n-1}+a_{n-2})/2$ and $a_1, a_2$ are given, will this series converge? And if so, what is the limit? By intuition I think it converges to $(a_1+2a_2)/3$ , but I am not able to prove it.

Question 12

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.

Question 13

How do you find an algebraic formula for $\sum_{k=1}^n k^3$? I am able to find one for $\sum_{k=1}^n k^2$, but not $k^3$. Any hints would be appreciated.

Question 14

Say you picked your favorite irrational number $q$ and looking at $S = \{nq: n\in \mathbb{Z} \}$ in $\mathbb{R}$, you chopped off everything but the decimal of $nq$, leaving you with a number in $[0,1]...

Question 15

Recently, I ran across a product that seems interesting. Does anyone know how to get to the closed form: $$\prod_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)=-\frac{\sin(\frac{n\pi}{2})}{2^{n-1}}$$ I ...

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All Questions

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Does an increasing sequence of reals converge if the difference of consecutive terms approaches zero?

Given $\langle f_{n}\rangle$ such that $f_{n}=\frac{f_{n-1}+f_{n-2}}{2}$ $\forall n\gt2$,to prove it converges to $\frac{f_1+2f_2}{3}$

Proving a general rule which states where a recursive series converges

Limit of a mean sequence

Closed form for $\sum_{n=0}^{\infty} \binom{n+k}{k} x^n$ (Negative Binomial Theorem)

Proof of convergence of a recursive sequence

How To Prove:$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4} = -\frac{7}{{720}}{\pi ^4}$

Will this series converge? If so, what is its limit?

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

How to determine equation for $\sum_{k=1}^n k^3$

Multiples of an irrational number forming a dense subset

Evaluating the product $\prod\limits_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)$

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