<s>[INST] What is $\sum_n n^6$ series? I went to OEIS but I didn't quite understand the formula for sum of this series. Could someone type it in normal shape and answer me?
and if it's no bother please do this for 7th, 8th, 9th and 10th powers?
 [/INST] $$
\sum_{k=1}^n k^6 = \frac{1}{42} n (n+1) (2 n+1) (3 n^4+6 n^3-3 n+1)\\
\sum_{k=1}^n k^7 = \frac{1}{24} n^2 (n+1)^2 (3 n^4+6 n^3-n^2-4 n+2)\\
\sum_{k=1}^n k^8 = \frac{1}{90} n (10 n^8+45 n^7+60 n^6-42 n^4+20 n^2-3)\\
\sum_{k=1}^n k^9 = \frac{1}{20} n^2 (n+1)^2 (n^2+n-1) (2 n^4+4 n^3-n^2-3 n+3)\\
\sum_{k=1}^n k^{10} = \frac{1}{66} n (6 n^{10}+33 n^9+55 n^8-66 n^6+66 n^4-33 n^2+5)
$$
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