go with the flow

February 12, 2010 in Lectures Without Words

s:=
[1^2, 2^2, 3^2, 4^2, ...]=
[1,4,9,16,...]=
[n^2]_(n \in \Bbb N).

=: “\Bbb N”
NB: 0 \in \Bbb N.
******************************************

S := x + 4x^2 + 9x^3 + …

()
******************************************
(1-x)*S = x + 3x^2 + 5x^3 + 7x^4 + …
(1-x)^2*S = x + 2x^2 + 2x^3 + 2x^4 + …
(1-x)^3*S = x + x^2

S = (x+x^2)/(1-x)^3
******************************************
(
x = 1/10

S = .1 + 4*.01 + 9*.001 + 16*.0001 + …
S ~ .150892
S = (.1+.01)/(.9)^3
S ~ .150892
OK
)

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