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'sFormula and Generating FunctionsQ3)𝑎𝑛=6𝑎𝑛−1−8𝑎𝑛−2𝑎𝑛+2=6𝑎𝑛+1−8𝑎𝑛∑𝑎𝑛+2𝑥𝑛∞𝑛=0=6∑𝑎𝑛+1𝑥𝑛∞𝑛=0−8∑𝑎𝑛𝑥𝑛∞𝑛=0∑𝑎𝑛𝑥𝑛−2∞𝑛=2=6∑𝑎𝑛𝑥𝑛−1∞𝑛=1−8∑𝑎𝑛𝑥𝑛∞𝑛=0∑𝑎𝑛𝑥²𝑥𝑛∞𝑛=2=6∑𝑎𝑛𝑥𝑥𝑛∞𝑛=1−8∑𝑎𝑛𝑥𝑛∞𝑛=0Therefore if we have(𝑥)=∑𝑎𝑛𝑥𝑛∞𝑛=0:1𝑥²∑𝑎𝑛𝑥𝑛∞𝑛=0−𝑎0𝑥²−𝑎1𝑥𝑥²=6𝑥∑𝑎𝑛𝑥𝑛∞𝑛=0−6𝑎0𝑥−8∑𝑎𝑛𝑥𝑛∞𝑛=01𝑥²𝑓−1𝑥²−2𝑥=6𝑥𝑓−6𝑥−8𝑓Therefore:(1𝑥²−6𝑥+8)𝑓=1𝑥²+2𝑥−6𝑥=1𝑥2−4𝑥𝑓(𝑥)=1𝑥2−4𝑥1𝑥²−6𝑥+8=1−4𝑥1−6𝑥+8𝑥²Q4)05+15+⋯+𝑛5=∑𝑘5𝑛𝑘=0This sum can be approximated by the Faulhaber formula :∑𝑘𝑝=1𝑝+1(𝑛𝑝+1+12(𝑝+1)𝑛𝑝+16(𝑝+12)𝑛𝑝−1−130(𝑝+14)𝑛𝑝−3+142(𝑝+16)𝑛𝑝−5𝑛𝑘=1+⋯)In this case p = 5:

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