query: How do I prove that $1^4+2^4+3^4\cdots\ + n^4 = \frac{1}{5}n^5 + \frac{1}{2}n^4 + \frac{1}{3}n^3-\frac{1}{30}n$?
positive_passages.docid: 158004
positive_passages.text: A formula for $1^4+2^4+...+n^4$
negative_passages.docid: 153594
negative_passages.text: When is $1^5 + 2^5 + \ldots + n^5$ a square?
negative_passages.docid: 50599
negative_passages.text: Prove that $\frac{1}{1*3}+\frac{1}{3*5}+\frac{1}{5*7}+...+\frac{1}{(2n-1)(2n+1)}=\frac{n}{2n+1}$
negative_passages.docid: 23726
negative_passages.text: A closed form for $1^{2}-2^{2}+3^{2}-4^{2}+ \cdots + (-1)^{n-1}n^{2}$
negative_passages.docid: 91068
negative_passages.text: Proving that $f_2+f_4+\cdots+f_{2n}=f_{2n+1}-1$ for Fibonacci numbers by induction
negative_passages.docid: 55481
negative_passages.text: Prove the identity $\binom{2n+1}{0} + \binom{2n+1}{1} + \cdots + \binom{2n+1}{n} = 4^n$
negative_passages.docid: 13902
negative_passages.text: Show that $\frac{1}{x_1^2}+\frac{1}{4\cdot x_2^2}+\ldots+\frac{1}{n^2\cdot x_n^2}\leq\frac{3n-2}{2n-1}$.
negative_passages.docid: 44247
negative_passages.text: For $a^3+b^3+c^3=3$ prove that $a^4b+b^4c+c^4a\leq3$
negative_passages.docid: 123820
negative_passages.text: Guess the formula for $\sum\frac 1{(4n-3)(4n+1)}$ and prove by induction
negative_passages.docid: 105238
negative_passages.text: Proving $\tan(\theta_1+\theta_2+\cdots+\theta_n)$ has the form $\frac{S_1-S_3+S_5+\cdots}{1-S_2+S_4+\cdots}$
negative_passages.docid: 78985
negative_passages.text: Prove that: $\sqrt{2\sqrt{3\sqrt{4\cdots\sqrt{n}}}}<3,\,\forall n\in\mathbb N.$
negative_passages.docid: 14306
negative_passages.text: What is the formula for $\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n(n+1)}$
negative_passages.docid: 74898
negative_passages.text: $1^n +2^n + \cdots +(p-1)^n \mod p =$?
negative_passages.docid: 80216
negative_passages.text: Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$
negative_passages.docid: 86900
negative_passages.text: Formula for the $1\cdot 2 + 2\cdot 3 + 3\cdot 4+\ldots + n\cdot (n+1)$ sum
negative_passages.docid: 142992
negative_passages.text: Prove that for every positive integer $n$, $1/1^2+1/2^2+1/3^2+\cdots+1/n^2\le2-1/n$
negative_passages.docid: 114074
negative_passages.text: Intuitive/Visual proof that $(1+2+\cdots+n)^2=1^3+2^3+\cdots+n^3$