# General way to find $$\sum _{ x=1 }^{ i }{ { x }^{ n } } ,\quad n\in N$$.

Let $$\sum _{ x=1 }^{ i }{ { x }^{ n } } ,\quad n\in N=f(n)$$.

Using the following fact and a bit of working around we can find f(n).

That is, $(1+x)^\alpha = \sum_{k=0}^{\alpha} \binom{\alpha}{k} x^k$ , where $\binom{\alpha}{k} = \frac{\alpha(\alpha-1)\cdots(\alpha-k+1)}{k!}.$

## For finding f(n) we must know f(1),f(2),......f(n-1).

The following method illustrates the way to find the sum of 4th power of natural numbers , the same can be used for finding for any nth power.

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## To try a problem based on this go below.

Find the area bounded between $$y = \lfloor x\rfloor ^4$$, the $$x$$-axis, $$x = 0$$ and $$x=11$$.

## Notation: $$\lfloor \cdot \rfloor$$ denotes the floor function.

Note by Sanath Balaji
1 year, 3 months ago

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There's another method also known as Faulhaber's Formula.

- 1 month, 3 weeks ago