# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ?
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### 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 $c_0 = {\alpha}^{\beta}$.
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### 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.
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### 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 ?
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### 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 ...
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### 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 ...
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): ...
### 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 \$\ \ \ \ \ \ \ ...