let c = 1⁴ + 2⁴ + 3⁴ + 4⁴ + 5⁴….
thus 16c = 2⁴ + 4⁴ + 6⁴ + 8⁴…..
thus c – 32c = -31c = 1⁴ – 2⁴ + 3⁴ – 4⁴ …..
thus this can be simplified to:
-3(2² + 1²) – 7(3² + 4²) -11(5² + 6²)…
which can be equivalent to:
-3(Σ(n²)) – 4( 3² +4² + 2(5² + 6²) + 3(7² + 8²)
which can be written as:
-3(Σ(n²)) – 4( 3² +4² + 5² + 6²… + (5² + 6²…..) + (7² + 8²…..)…..)
which is equivalent to:
-3(Σ(n²)) – 4( ∞Σ(n²) – ∞Σ(n²))
which is simplified down to:
-3(Σ(n²))
thus -31c = 15/168 which is equal to 5/56 thus c = -5/1736
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