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The Summation of 1/4ⁿ, as "n" goes from 1 to infinity is 1/3 Visual proof

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every the addition aggregate sum part accretion accruing adding computing counting enlarging expanding increasing reckoning summation summing tabulating totaling toting decrease deduction lessening loss reduction shrinkage subtraction withdrawal calculation arithmetic ciphering computation dividing estimate estimation figuring forecast judgment multiplying prediction subtracting calculations addings arithmetics cipherings computations countings dividings estimates estimations figurings forecasts judgments multiplyings predictions reckonings subtractings summations totalings conspectus abstract digest outline overview précis resume review rundown résumé synopsis ending break breakoff catastrophe cessation close closing closure completion consummation coup de grace culmination denouement desistance dissolution epilogue expiration finale finish lapse omega outcome period resolution stop swan song termination terminus upshot wane wind-up beginning opening start commencement initiation introduction success afterword back matter coda concluding speech conclusion follow-up peroration postlude postscript sequel foreword preface activity animation bang birr drive energy force get-up-and-go hardihood life moxie oomph pep potency push snap starch tuck verve vigor vivacity zest apathy discouragement dullness idleness inactivity laziness lethargy weakness essay whirl shot fling bid pop turn stab slap crack effort whack fly retire travel move leave pass progress flee fare repair hightail lam depart quit wend escape mosey near journey exit hie withdraw cruise decamp approach vamoose abscond split beat it get off hit the road light out pull out push off shove off take off bug out get away get going get lost make a break for it make for move out push on run along run away set off skip out take a hike take a powder take flight take leave make one's way be born mismatch disagree break continue enter come arrive stop stay face surrender create wait remain work score run prosper perform persist maintain succeed act flourish click thrive carry on make out pan out give up lose fail halt languish cease extend lead make reach range vary spread cover connect fit give access obey develop proceed serve happen persevere concur incline conduce transpire befall eventuate tend avail chance result wax occur fall out turn out work out lead to cause set like enjoy conform accord square dovetail complement belong correspond mesh jibe suit chime blend relish match be adapted for be designed for differ disconnect refuse hate dislike finish give drop spend decease consume conclude terminate crumble deplete bend squander decline worsen cave dissipate weaken succumb exhaust perish devour demise expend waste yield expire fritter pass away run through fold up pass on use up hold save start begin build gather collect bear earn keep improve hoard increase commence store accumulate lapse flow be spent slip away waste away take allow stand stomach tolerate brook let abide permit swallow suffer consent to put up with shun disallow reject against in distinction to out of possession of taken away beginning at coming out of deriving out of originating at starting with minus defective diminished from lacking less negative subtracted without plus native aboriginal autochthonous belonging endemic homegrown homemade indigenous local national original primary primeval primitive regional foreign alien outside about concerning like regarding since after afterward ago already because considering for hence later now past subsequently therefore whereas boundlessness continuity continuum eternity expanse extent immensity infinitude limitlessness myriad perpetuity sempiternity space ubiquity vastness beyond endless time immeasurability unlimited space vastitude bounds definiteness ending limitation eternality ages and ages ceaselessness eternalness everlastingness forever infinity permanence perpetuation timelessness eternities aeons afterlifes ages blue moons dog's ages endless times endlessness everlastingnesses forever and a days futures immortality imperishabilities infinitenesses infinities kingdom comes other worlds time without ends wild blue yonders world without ends aeon afterlife age all time blue moon dog's age forever and a day future imperishability infiniteness kingdom come other world time without end wild blue yonder world without end aftertime afterward by-and-by destiny expectation fate futurity hereafter life to come millennium morrow offing outlook posterity prospect subsequent time to-be tomorrow world to come past greatness abundance amplitude bigness bulk enormity force high degree hugeness intensity length magnitude mass might potency power prodigiousness sizableness strength lethargy weakness inability incompetence insignificance powerlessness smallness tininess unimportance ocular optic optical beheld discernible imaged observable observed perceptible seeable seen viewable viewed visible visional graphic blocked-out delineated depicted descriptive diagrammatic drawn engraved etched iconographic illustrated illustrational illustrative marked-out outlined painted photographic pictoric pictured portrayed representational sketched traced visual implicit obscure unclear vague model cartoon clone copy copycat dead ringer ditto dummy duplicate effigy engraving facsimile figure figurine game plan illustration image knockoff layout look-alike miniature mock-up painting paste-up photograph picture pocket portrait print relief representation ringer setup sketch spitting image statue statuette tracing being entity original photographer models configurations designs forms kinds marks modes styles varieties photographers more visible arresting big as life bold clear conspicuous detectable discoverable distinguishable evident in sight in view inescapable macroscopic manifest marked not hidden noticeable obtrusive obvious open out in the open outstanding palpable patent perceivable plain pointed pronounced revealed salient signal striking to be seen unconcealed under one's nose unhidden unmistakable unseeable secret unnoticeable invisible concealed ambiguous inconspicuous unsure unknown uncertain indefinite hidden obscured indistinct eye ophthalmic sight seeing argument clue confirmation criterion data demonstration information picture reason testament testimony validation verification affidavit attestation averment case certification clincher corroboration credentials cue deposition documents establishment exhibit facts goods grabber grounds nitty-gritty reasons record scoop score skinny substantiation trace warrant wherefore why witness chapter and verse lowdown paper trail smoking gun straight stuff whyfor hypothesis theory impression galley pass pull revise slip stereo trial galley proof page proof repro trial print trial proof acid test analysis conclusive test crucial test decisive test definitive test experiment genuineness proof proving ground sure test test of value alibi account affirmation airtight case allegation answer assertion assurance avowal cop-out cover declaration excuse explanation fish story justification plea pretext profession reply retort song and dance stall statement vindication denial question request argumentation claim contention debate defense discussion exchange expostulation line of reasoning logic pleading polemic questioning reasoning remonstrance remonstration arguments argumentations assertions cases claims contentions debates defenses discussions exchanges expostulations line of reasonings logics pleadings pleas polemics proofs questionings reasonings remonstrances remonstrations authentication documentation evidence testimonial
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