# Tagged Questions

For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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### Interesting sequence using limits

Can you please solve the problem I am stuck
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### Is there any formula to find the nth element in a sequence where common difference (d) is varying with a constant rate?

To explain my question, here is an example. Below is an AP: 2, 6, 10, 14....n Calculating the nth term in this sequence is easy because we have a formula. The common difference (d = 4) in AP is ...
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### Write an expression in terms of $n$ for the $n$th term in the following sequence: $9,16,25,36,49$

Write an expression in terms of $n$ for the $n$th term in the following sequence: $9,16,25,36,49$ The difference is $+ 7 , + 9 , + 11 , + 13, + 15 , + 17 ,$ etc The difference is not constant so ...
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### How can I find the limit of the $s_n=n[1+(-1)^n]$

$s_n=n[1+(-1)^n]$ Any hints on how to get started with this one?
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### I derived a new formula related to arithmetic sequences, I think!

First of all, I am a 12th student so I don't know how to write research notes. So please forgive me if my writing is not so impressive! I don't know what to do to let the world know about whatever I ...
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### Alternating sum of polygamma functions

I've come across the following sum while doing a problem: $$\sum_{n=0}^\infty (-1)^n \psi^{(1)}(m+np)$$ Where $m \in (0,\infty)$ and $p \in \mathbb{N}$ and $\psi^{(1)}$ denotes the first derivative ...
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### Does $a_n \sim b_n$ imply $\sum_n a_n \sim \sum_n b_n$ for $a_n, b_n>0$?

I am almost embarrassed to asked to this question, but after considering it for a while I realize I need some help. In the following $a_n, b_n >0$. So, by limit comparison test if $a_n = O(b(n))$ ...
301 views

### What is $\sum_{r=0}^n \frac{(-1)^r}{\binom{n}{r}}$?

Find a closed form expression for $$\sum_{r=0}^n \dfrac{(-1)^r}{\dbinom{n}{r}}$$ where $n$ is an even positive integer. I tried using binomial identities, but since the binomial coefficient ...
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### $x_{n+m}\le \frac{x_n+x_{n+1}+\cdots+x_{n+m-1}}{m}$. Prove that this sequence has a limit.

Let $m \ge 2 -$ fixed positive integer. The sequence of non-negative real numbers $\{x_n\}_{n=1}^{\infty}$ is that for all $n\in \mathbb N$ $$x_{n+m}\le \frac{x_n+x_{n+1}+\cdots+x_{n+m-1}}{m}$$ ...