Question-97564

Question Number 97564 by Power last updated on 08/Jun/20

Answered by Rio Michael last updated on 09/Jun/20

Ones our teacher put this on the board, we puzzeled even the geniuses of our class tried it out but it turned out even the teacher couldn′t find a particular summation formula which described this series even to an approximate. Some 1945 folks had tried this problem and concluded that it can only be programmed hence no exact summation formula exist for it. I tried it out and noticed(like others) that there is no formula to compute this sum! but we can obtain a bound to which this series falls : n(1 + (1/(4(n−1)))) < S < n^n (1 + (2/(e(n−1)))) please do well to search for the programming codes for this series.

$$\mathrm{Ones}\:\mathrm{our}\:\mathrm{teacher}\:\mathrm{put}\:\mathrm{this}\:\mathrm{on}\:\mathrm{the}\:\mathrm{board},\:\mathrm{we}\:\mathrm{puzzeled}\:\mathrm{even} \\ $$$$\mathrm{the}\:\mathrm{geniuses}\:\mathrm{of}\:\mathrm{our}\:\mathrm{class}\:\mathrm{tried}\:\mathrm{it}\:\mathrm{out}\:\mathrm{but}\:\mathrm{it}\:\mathrm{turned}\:\mathrm{out} \\ $$$$\mathrm{even}\:\mathrm{the}\:\mathrm{teacher}\:\mathrm{couldn}'\mathrm{t}\:\mathrm{find}\:\mathrm{a}\:\mathrm{particular}\:\mathrm{summation}\:\mathrm{formula} \\ $$$$\mathrm{which}\:\mathrm{described}\:\mathrm{this}\:\mathrm{series}\:\mathrm{even}\:\mathrm{to}\:\mathrm{an}\:\mathrm{approximate}.\:\mathrm{Some}\: \\ $$$$\mathrm{1945}\:\mathrm{folks}\:\mathrm{had}\:\mathrm{tried}\:\mathrm{this}\:\mathrm{problem}\:\mathrm{and}\:\mathrm{concluded}\:\mathrm{that}\:\mathrm{it}\:\mathrm{can}\:\mathrm{only} \\ $$$$\mathrm{be}\:\mathrm{programmed}\:\mathrm{hence}\:\mathrm{no}\:\mathrm{exact}\:\mathrm{summation}\:\mathrm{formula}\:\mathrm{exist}\:\mathrm{for}\:\mathrm{it}. \\ $$$$\mathrm{I}\:\mathrm{tried}\:\mathrm{it}\:\mathrm{out}\:\mathrm{and}\:\mathrm{noticed}\left(\mathrm{like}\:\mathrm{others}\right)\:\mathrm{that}\:\mathrm{there}\:\mathrm{is}\:\boldsymbol{\mathrm{no}}\:\boldsymbol{\mathrm{formula}} \\ $$$$\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{compute}}\:\boldsymbol{\mathrm{this}}\:\boldsymbol{\mathrm{sum}}! \\ $$$$\mathrm{but}\:\mathrm{we}\:\mathrm{can}\:\mathrm{obtain}\:\mathrm{a}\:\mathrm{bound}\:\mathrm{to}\:\mathrm{which}\:\mathrm{this}\:\mathrm{series}\:\mathrm{falls} \\ $$$$:\:{n}\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{4}\left({n}−\mathrm{1}\right)}\right)\:<\:{S}\:<\:{n}^{{n}} \left(\mathrm{1}\:+\:\frac{\mathrm{2}}{{e}\left({n}−\mathrm{1}\right)}\right) \\ $$$$\mathrm{please}\:\mathrm{do}\:\mathrm{well}\:\mathrm{to}\:\mathrm{search}\:\mathrm{for}\:\mathrm{the}\:\mathrm{programming}\:\mathrm{codes}\:\mathrm{for}\:\mathrm{this}\:\mathrm{series}. \\ $$

Commented by Power last updated on 09/Jun/20

$$\mathrm{thanks}\:\mathrm{sir} \\ $$

Facebook Tweet Pin

Leave a Reply Cancel reply