Question and Answers Forum

All Questions      Topic List

Geometry Questions

Previous in All Question      Next in All Question      

Previous in Geometry      Next in Geometry      

Question Number 2138 by Filup last updated on 06/Nov/15

$$\mathrm{Is}\mathrm{the}\mathrm{following}\mathrm{proof}\mathrm{correct}?$$$$$$$$\mathrm{\Delta }=\underset{i=0}{\sum ^{\mathrm{\infty }}}{(-1)}^i{2}^i=1-2+4-8+16-32+...$$$$$$$$\mathrm{Let}:$$$${\mathrm{\Delta }}_1=1-2+4-8+16-32+...$$$${\mathrm{\Delta }}_2=1-2+4-8+16-32+...$$$$$$$${\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2=1+(-2+1)+(4-2)+(-8+4)+...$$$$\therefore {\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2=1-(1-2+4-8+...)$$$$\therefore {\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2=1-{\mathrm{\Delta }}_1$$$${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=\mathrm{\Delta }$$$$3\mathrm{\Delta }=1$$$$\therefore \mathrm{\Delta }=\frac{1}{3}$$$$$$$$\therefore \mathrm{\Delta }=\underset{i=0}{\sum ^{\mathrm{\infty }}}{(-1)}^i{2}^i=\frac{1}{3}$$$$$$

Commented by prakash jain last updated on 04/Nov/15

$$\mathrm{You}\mathrm{can}\mathrm{NOT}\mathrm{add}2\mathrm{divergent}\mathrm{series}\mathrm{and}\mathrm{expect}$$$$\mathrm{reesult}\mathrm{to}\mathrm{be}\mathrm{double}.\mathrm{Same}\mathrm{for}\mathrm{subtraction}.$$

Commented by Filup last updated on 04/Nov/15

$$\mathrm{A}\mathrm{proof}\mathrm{of}1+2+3+4+5+...=-\frac{1}{12}$$$$\mathrm{involves}\mathrm{this}\mathrm{method},\mathrm{though}.$$$$$$$$\mathrm{Also},\mathrm{according}\mathrm{to}{\mathrm{wikipedia}}^{\prime }\mathrm{s}\mathrm{page}\mathrm{for}$$$$\mathrm{this}\mathrm{sequence},\mathrm{this}\mathrm{answer}\mathrm{is}\mathrm{correct}.$$

Commented by Filup last updated on 04/Nov/15

$$Zerosrepresentspaces$$$$S_1=1+2+3+4+0$$$$S_2=0+1+2+3+4$$$$$$$$\mathrm{I}\mathrm{am}\mathrm{usure}\mathrm{as}\mathrm{to}\mathrm{why}\mathrm{the}\mathrm{attempted}$$$$\mathrm{proof}\mathrm{is}\mathrm{incorrect}\mathrm{dispite}\mathrm{getting}\mathrm{the}$$$$\mathrm{same}\mathrm{result}\mathrm{as}\mathrm{even}\mathrm{Euler}$$$$$$

Commented by prakash jain last updated on 04/Nov/15

$$\mathrm{Ok}.\mathrm{You}\mathrm{are}\mathrm{talking}\mathrm{about}\mathrm{a}\mathrm{different}\mathrm{sum}.$$$$\mathrm{I}\mathrm{misunderstood}.$$

Commented by 123456 last updated on 04/Nov/15

$$\mathrm{other}\mathrm{sumations}\mathrm{methods}\mathrm{and}\mathrm{analitic}\mathrm{clntinuation}$$

Commented by Filup last updated on 04/Nov/15

$$\mathrm{Because}\mathrm{i}\mathrm{am}\mathrm{only}\mathrm{an}\mathrm{amature}\mathrm{mathematian},$$$$\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{really}\mathrm{understand}\mathrm{what}\mathrm{it}\mathrm{is}$$$$\mathrm{I}\mathrm{have}\mathrm{done}.\mathrm{Please}\mathrm{explain}\mathrm{how}\mathrm{this}\mathrm{is}$$$$\mathrm{a}\mathrm{different}\mathrm{type}\mathrm{of}\mathrm{sum}?$$

Commented by prakash jain last updated on 04/Nov/15

$$\mathrm{My}\mathrm{mistake}.\mathrm{I}\mathrm{misread}\mathrm{your}\mathrm{question}.\mathrm{Also}$$$$\mathrm{ignore}\mathrm{my}\mathrm{previous}\mathrm{comment}\mathrm{about}$$$$\mathrm{divergent}\mathrm{series}.$$

Commented by Filup last updated on 04/Nov/15

$$\mathrm{Ok}\mathrm{thank}\mathrm{you}\mathrm{very}\mathrm{much}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com