Advanced Series And Recurrence Relations: Applications Of Faulhaber's Formula And Generating Functions

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Advanced Series and Recurrence Relations: Applications of Faulhaber's Formula and Generating Functions Q3) 𝑎 𝑛 = 6 𝑎 𝑛 − 1 − 8 𝑎 𝑛 − 2 𝑎 𝑛 + 2 = 6 𝑎 𝑛 + 1 − 8 𝑎 𝑛 ∑ 𝑎 𝑛 + 2 𝑥 𝑛 ∞ 𝑛 = 0 = 6 ∑ 𝑎 𝑛 + 1 𝑥 𝑛 ∞ 𝑛 = 0 − 8 ∑ 𝑎 𝑛 𝑥 𝑛 ∞ 𝑛 = 0 ∑ 𝑎 𝑛 𝑥 𝑛 − 2 ∞ 𝑛 = 2 = 6 ∑ 𝑎 𝑛 𝑥 𝑛 − 1 ∞ 𝑛 = 1 − 8 ∑ 𝑎 𝑛 𝑥 𝑛 ∞ 𝑛 = 0 ∑ 𝑎 𝑛 𝑥 ² 𝑥 𝑛 ∞ 𝑛 = 2 = 6 ∑ 𝑎 𝑛 𝑥 𝑥 𝑛 ∞ 𝑛 = 1 − 8 ∑ 𝑎 𝑛 𝑥 𝑛 ∞ 𝑛 = 0 Therefore if we have ( 𝑥 ) = ∑ 𝑎 𝑛 𝑥 𝑛 ∞ 𝑛 = 0 : 1 𝑥 ² ∑ 𝑎 𝑛 𝑥 𝑛 ∞ 𝑛 = 0 − 𝑎 0 𝑥 ² − 𝑎 1 𝑥 𝑥 ² = 6 𝑥 ∑ 𝑎 𝑛 𝑥 𝑛 ∞ 𝑛 = 0 − 6 𝑎 0 𝑥 − 8 ∑ 𝑎 𝑛 𝑥 𝑛 ∞ 𝑛 = 0 1 𝑥 ² 𝑓 − 1 𝑥 ² − 2 𝑥 = 6 𝑥 𝑓 − 6 𝑥 − 8 𝑓 Therefore : ( 1 𝑥 ² − 6 𝑥 + 8 ) 𝑓 = 1 𝑥 ² + 2 𝑥 − 6 𝑥 = 1 𝑥 2 − 4 𝑥 𝑓 ( 𝑥 ) = 1 𝑥 2 − 4 𝑥 1 𝑥 ² − 6 𝑥 + 8 = 1 − 4 𝑥 1 − 6 𝑥 + 8 𝑥 ² Q4) 0 5 + 1 5 + ⋯ + 𝑛 5 = ∑ 𝑘 5 𝑛 𝑘 = 0 This sum can be approximated by the Faulhaber formula : ∑ 𝑘 𝑝 = 1 𝑝 + 1 ( 𝑛 𝑝 + 1 + 1 2 ( 𝑝 + 1 ) 𝑛 𝑝 + 1 6 ( 𝑝 + 1 2 ) 𝑛 𝑝 − 1 − 1 30 ( 𝑝 + 1 4 ) 𝑛 𝑝 − 3 + 1 42 ( 𝑝 + 1 6 ) 𝑛 𝑝 − 5 𝑛 𝑘 = 1 + ⋯ ) In this case p = 5 :
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