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

Simplify recurrence relations with this expert Assignment Solution!

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Advanced Series and Recurrence Relations: Applications of Faulhaber'sFormula and Generating FunctionsQ3)π‘Žπ‘›=6π‘Žπ‘›βˆ’1βˆ’8π‘Žπ‘›βˆ’2π‘Žπ‘›+2=6π‘Žπ‘›+1βˆ’8π‘Žπ‘›βˆ‘π‘Žπ‘›+2π‘₯π‘›βˆžπ‘›=0=6βˆ‘π‘Žπ‘›+1π‘₯π‘›βˆžπ‘›=0βˆ’8βˆ‘π‘Žπ‘›π‘₯π‘›βˆžπ‘›=0βˆ‘π‘Žπ‘›π‘₯π‘›βˆ’2βˆžπ‘›=2=6βˆ‘π‘Žπ‘›π‘₯π‘›βˆ’1βˆžπ‘›=1βˆ’8βˆ‘π‘Žπ‘›π‘₯π‘›βˆžπ‘›=0βˆ‘π‘Žπ‘›π‘₯Β²π‘₯π‘›βˆžπ‘›=2=6βˆ‘π‘Žπ‘›π‘₯π‘₯π‘›βˆžπ‘›=1βˆ’8βˆ‘π‘Žπ‘›π‘₯π‘›βˆžπ‘›=0Therefore if we have(π‘₯)=βˆ‘π‘Žπ‘›π‘₯π‘›βˆžπ‘›=0:1π‘₯Β²βˆ‘π‘Žπ‘›π‘₯π‘›βˆžπ‘›=0βˆ’π‘Ž0π‘₯Β²βˆ’π‘Ž1π‘₯π‘₯Β²=6π‘₯βˆ‘π‘Žπ‘›π‘₯π‘›βˆžπ‘›=0βˆ’6π‘Ž0π‘₯βˆ’8βˆ‘π‘Žπ‘›π‘₯π‘›βˆžπ‘›=01π‘₯Β²π‘“βˆ’1π‘₯Β²βˆ’2π‘₯=6π‘₯π‘“βˆ’6π‘₯βˆ’8𝑓Therefore:(1π‘₯Β²βˆ’6π‘₯+8)𝑓=1π‘₯Β²+2π‘₯βˆ’6π‘₯=1π‘₯2βˆ’4π‘₯𝑓(π‘₯)=1π‘₯2βˆ’4π‘₯1π‘₯Β²βˆ’6π‘₯+8=1βˆ’4π‘₯1βˆ’6π‘₯+8π‘₯Β²Q4)05+15+β‹―+𝑛5=βˆ‘π‘˜5π‘›π‘˜=0This sum can be approximated by the Faulhaber formula :βˆ‘π‘˜π‘=1𝑝+1(𝑛𝑝+1+12(𝑝+1)𝑛𝑝+16(𝑝+12)π‘›π‘βˆ’1βˆ’130(𝑝+14)π‘›π‘βˆ’3+142(𝑝+16)π‘›π‘βˆ’5π‘›π‘˜=1+β‹―)In this case p = 5:

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