Difference between revisions of "Proofs without words"
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== Summations == | == Summations == | ||
− | <center><asy>defaultpen(linewidth(0.7)); unitsize(15); pen | + | <center><asy>unitsize(15); defaultpen(linewidth(0.7)); |
+ | int n = 6; pair shiftR = ((n+2),0); real r = 0.3; | ||
+ | pen colors(int i){ return rgb(i/n,0.4+i/(2n),1-i/n); } /* shading */ | ||
+ | void htick(pair A, pair B,pair ticklength = (0.15,0)){ draw(A--B ^^ A-ticklength--A+ticklength ^^ B-ticklength--B+ticklength); } | ||
+ | |||
+ | /* triangle */ | ||
+ | draw((-r,0)--(-r,-n+1)^^(r,-n+1)--(r,0),linetype("4 4")); | ||
+ | for(int i = 0; i < n; ++i) | ||
+ | draw((-i,-i)--(i,-i)); | ||
+ | for(int i = 0; i < n; ++i) | ||
+ | for(int j = 0; j < 2*i+1; ++j) | ||
+ | filldraw(CR((j-i,-i),r),colors(i)); | ||
+ | |||
+ | /* square */ | ||
+ | draw(r*expi(pi/4)+shiftR--(n-1,-n+1)+r*expi(pi/4)+shiftR^^r*expi(5*pi/4)+shiftR--r*expi(5*pi/4)+(n-1,-n+1)+shiftR,linetype("4 4")); | ||
+ | for(int i = 0; i < n; ++i) | ||
+ | draw(shiftR+(0,-i)--shiftR+(i,-i)--shiftR+(i,0)); | ||
+ | for(int i = 0; i < n; ++i) | ||
+ | for(int j = 0; j < n; ++j) | ||
+ | filldraw(CR((j,-i)+shiftR,r),colors((i>j)?i:j)); | ||
+ | |||
+ | htick(shiftR+(-1,r),shiftR+(-1,-n+1-r)); label("$n$",shiftR+(-1,(-n+1)/2),W,fontsize(10)); | ||
+ | </asy><br> | ||
+ | |||
+ | The sum of the first <math>n</math> odd natural numbers is <math>n^2</math>.<br><br> | ||
+ | </center> | ||
+ | |||
+ | <center><asy> defaultpen(linewidth(0.7)); unitsize(15); | ||
+ | int n = 6; pair shiftR = ((n+2),0); real r = 0.3; | ||
+ | pen colors(int i){ return rgb(0.4+i/(2n),i/n,1-i/n); } /* shading */ | ||
+ | void htick(pair A, pair B,pair ticklength = (0.15,0)){ | ||
+ | draw(A--B); | ||
+ | draw(A-ticklength--A+ticklength); | ||
+ | draw(B-ticklength--B+ticklength); | ||
+ | } | ||
+ | |||
+ | /* triangle */ | ||
+ | draw((0.5,0)--(n-0.5,-n+1),linetype("4 4")); | ||
+ | for(int i = 0; i < n; ++i) | ||
+ | draw((0,-i)--(i,-i)); | ||
+ | for(int i = 0; i < n; ++i) | ||
+ | for(int j = 0; j <= i; ++j) | ||
+ | filldraw(CR((j,-i),r),colors(i)); | ||
+ | |||
+ | /* arc arrow */ | ||
+ | draw( arc((n,-n+1)/2, (1.5,-1.5), (n-1.5,-1.5), CW) ); | ||
+ | fill((n-1.5,-1.5) -- (n-1.5,-1.5)+r*expi(5.2*pi/6) -- (n-1.5,-1.5)+r*expi(3.3*pi/6) -- cycle); /* manual arrowhead? avoid resizing */ | ||
+ | |||
+ | /* square */ | ||
+ | draw(shiftR+(0.5,0)--shiftR+(n-0.5,-n+1),linetype("4 4")); | ||
+ | for(int i = 0; i < n; ++i) | ||
+ | draw(shiftR+(0,-i)--shiftR+(i,-i)^^shiftR+(n,-n+1)-(0,-i)--shiftR+(n,-n+1)-(i,-i)); | ||
+ | for(int i = 0; i < n; ++i) | ||
+ | for(int j = 0; j < n+1; ++j) | ||
+ | filldraw(CR((j,-i)+shiftR,r),colors((j <= i) ? i : n-1-i)); | ||
+ | |||
+ | /* labeling and ticks */ | ||
+ | htick(shiftR+(-1,r),shiftR+(-1,-n+1-r)); label("$n$",shiftR+(-1,(-n+1)/2),W,fontsize(10)); | ||
+ | htick(shiftR+(-r,-n),shiftR+(n+r-1,-n),(0,0.15)); label("$n$",shiftR+((n-1)/2,-n),S,fontsize(10)); | ||
+ | htick(shiftR+(n-r,-n),shiftR+(n+r,-n),(0,0.15)); label("$1$",shiftR+(n,-n),S,fontsize(10)); | ||
+ | </asy><br> | ||
+ | |||
+ | The sum of the first <math>n</math> positive integers is <math>n(n+1)/2</math>.<br><br> | ||
+ | </center> | ||
+ | |||
+ | <center><asy>unitsize(15); defaultpen(linewidth(0.7)); | ||
+ | int n = 6; real r = 0.35, h = 3/4; /* radius size and horizontal spacing */ | ||
+ | pair shiftR = (h*(n+1)+r, 0); | ||
+ | |||
+ | pen colors(int i){ /* shading */ | ||
+ | if(i == n) return red; | ||
+ | return rgb(5/n,0.4+5/(2n),1-5/n); | ||
+ | } | ||
+ | void htick(pair A, pair B, pair ticklength = (0.15,0)){ | ||
+ | draw(A--B ^^ A-ticklength--A+ticklength ^^ B-ticklength--B+ticklength); | ||
+ | } | ||
+ | void makeshiftarrow(pair A, real dir, real arrowlength = r){ /* Arrow option resizes */ | ||
+ | fill(A--A+arrowlength*expi(dir+pi/8)--A+arrowlength*expi(dir-pi/8)--cycle); | ||
+ | } | ||
+ | pair getCenter(int i, int j){ return ((2*j-i)*h,-i);} | ||
+ | |||
+ | /* triangle */ | ||
+ | for(int i = 0; i < n+1; ++i){ | ||
+ | draw((-i*h,-i)--(i*h,-i)); /* horizontal lining */ | ||
+ | for(int j = 0; j <= i; ++j) | ||
+ | filldraw(circle(getCenter(i,j),r), colors(i)); | ||
+ | } | ||
+ | |||
+ | /* fill in circle in row 4, column 3 */ | ||
+ | filldraw(circle(getCenter(3,2),r),blue); | ||
+ | draw(getCenter(n,2)-- getCenter(3,2)-- getCenter(n,n+2-3)); | ||
+ | makeshiftarrow(getCenter(n,2),pi/4,0.5); makeshiftarrow(getCenter(n,n+2-3),3*pi/4,0.5); | ||
− | + | htick(shiftR+(-1,r),shiftR+(-1,-n+1-r)); label("$n$",shiftR+(-1,(-n+1)/2),E,fontsize(10)); | |
− | + | </asy><br> | |
− | |||
− | |||
− | |||
− | + | The sum of the first <math>n</math> positive integers is <math>{n+1 \choose 2}</math>.{{ref|1}}<br><br> | |
+ | </center> | ||
− | + | <center><asy>defaultpen(linewidth(0.7)); unitsize(15); pen heavy = linewidth(2); | |
+ | int n2 = 4, n = floor(n2*(n2+1)/2); real h = 0.6; pair shiftR1 = (n*h+1,0), shiftR2 = shiftR1 + (n*h+1,0); /* global configurable variables */ | ||
− | /* | + | int lvl(int i){ return ceil(((8*i+9)^.5-1)/2); } |
− | void htick(pair A, pair B,pair ticklength = (0.15,0)) { draw(A--B ^^ A-ticklength--A+ticklength ^^ B-ticklength--B+ticklength); } | + | pen colors(int i){ return rgb(0.2+lvl(i)/6,0.3+lvl(i)/7,1-lvl(i)/6); } /* shading */ |
+ | void htick(pair A, pair B,pair ticklength = (0.15,0)){ draw(A--B ^^ A-ticklength--A+ticklength ^^ B-ticklength--B+ticklength); } | ||
/* gradient triangle */ | /* gradient triangle */ | ||
for(int i = 0; i < n; ++i){ | for(int i = 0; i < n; ++i){ | ||
for(int j = 0; j < 2*i+1; ++j){ | for(int j = 0; j < 2*i+1; ++j){ | ||
− | filldraw(shift(shiftR1)* | + | filldraw(shift(shiftR1)*xscale(h)*yscale(h)*shift((j-i,-i))*unitsquare,colors(i)); |
− | |||
if(j % lvl(i) == 0 && j != lvl(i)^2) | if(j % lvl(i) == 0 && j != lvl(i)^2) | ||
− | draw(shift(shiftR1)* | + | draw(shift(shiftR1)*xscale(h)*yscale(h)*shift((j-i,-i))*((0,0)--(0,1)--(1,1)), heavy); |
− | if(j == 2*i) / | + | if(j == 2*i) /* right border */ |
− | draw(shift(shiftR1)* | + | draw(shift(shiftR1)*xscale(h)*yscale(h)*shift((j-i,-i))*((1,0)--(1,1)--(0,1)), heavy); |
− | |||
} | } | ||
− | |||
− | |||
} | } | ||
− | + | for(int i = 0; i < n2; ++i) | |
− | + | draw(shift(shiftR1)*xscale(h)*yscale(h)*shift((-i*(i+1)/2,-i*(i+1)/2))*((0,1)--(2*i*(i+1)/2+1,1)), heavy); | |
− | + | draw(shift(shiftR1)*xscale(h)*yscale(h)*shift((-n2*(n2+1)/2,-n2*(n2+1)/2))*((1,1)--(2*n2*(n2+1)/2,1)), heavy); | |
− | for(int i = 0; i < n2; ++i) | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
/* gradient square */ | /* gradient square */ | ||
− | for(int i = 0; i < n; ++i) | + | for(int i = 0; i < n; ++i) |
for(int j = 0; j < n; ++j) | for(int j = 0; j < n; ++j) | ||
− | filldraw(shift(shiftR2)* | + | filldraw(shift(shiftR2)*xscale(h)*yscale(h)*shift((j,-i))*unitsquare,colors((i>j)?i:j)); |
− | |||
− | |||
− | |||
− | |||
− | |||
/* n nxn squares */ | /* n nxn squares */ | ||
for(int i = 0; i < n2; ++i){ | for(int i = 0; i < n2; ++i){ | ||
− | filldraw( | + | filldraw(xscale(h)*yscale(h)*shift((-i,-(i+1)*(i+2)/2+1))*xscale(i+1)*yscale(i+1)*unitsquare, colors(floor(i*(i+1)/2)), heavy); |
} | } | ||
</asy><br> | </asy><br> | ||
Line 123: | Line 190: | ||
Another proof of the identity <math>1^3 + 2^3 + \cdots + n^3 = (1+2+\cdots + n)^2</math>.<br><br></center> | Another proof of the identity <math>1^3 + 2^3 + \cdots + n^3 = (1+2+\cdots + n)^2</math>.<br><br></center> | ||
+ | |||
+ | <center><asy>defaultpen(linewidth(0.7)); unitsize(15); pen sm = fontsize(10); | ||
+ | int n = 5, fib = 1, fib2 = 1, xsum = 1, ysum = 0; real h = 0.15; | ||
+ | void fillsq(pair A = (0,0), real s, pen p = invisible, pen l = linewidth(1)){ filldraw(shift(A)*xscale(s)*yscale(s)*unitsquare, p, l); } | ||
+ | void htick(pair A, pair B, pair ticklength = (0.15,0)){ draw(A--B ^^ A-ticklength--A+ticklength ^^ B-ticklength--B+ticklength); } | ||
+ | |||
+ | for(int i = 0; i < n; ++i) { | ||
+ | fillsq((0,h*ysum),h*fib2,rgb(0.9,1,0.9)); | ||
+ | fillsq((h*xsum,0),h*fib,rgb(1,0.9,0.9)); | ||
+ | if(i == n-1){ | ||
+ | label("$F_{n}^2$",h*(xsum+fib/2,fib/2),sm); | ||
+ | label("$F_{n-1}^2$",h*(fib2/2,ysum+fib2/2),sm); | ||
+ | } | ||
+ | else if(i == n-2){ | ||
+ | label("$F_{n-2}^2$",h*(xsum+fib/2,fib/2),sm); | ||
+ | label("$F_{n-3}^2$",h*(fib2/2,ysum+fib2/2),sm); | ||
+ | } | ||
+ | fib = fib + fib2; fib2 = fib - fib2; | ||
+ | xsum = fib; | ||
+ | ysum = fib2; | ||
+ | fib = fib + fib2; fib2 = fib - fib2; | ||
+ | } | ||
+ | htick(h*(xsum,0)+(1,0),h*(xsum,ysum)+(1,0)); label("$F_n$",h*(xsum,ysum/2)+(1,0), E, sm); | ||
+ | htick(h*(0,ysum)+(0,1),h*(xsum-fib+fib2,ysum)+(0,1),(0,0.15)); label("$F_{n-1}$",h*((xsum-fib+fib2)/2,ysum)+(0,1), N, sm); | ||
+ | htick(h*(xsum,ysum)+(0,1),h*(xsum-fib+fib2,ysum)+(0,1),(0,0.15)); label("$F_{n}$",h*((2*xsum-fib+fib2)/2,ysum)+(0,1), N, sm); | ||
+ | </asy><br> | ||
+ | |||
+ | The identity <math>F_1^2 + F_2^2 + \cdots + F_n^2 = F_{n} \cdot F_{n+1}</math>, where <math>F_i</math> is the <math>i</math>th [[Fibonacci number]].<br><br></center> | ||
+ | |||
+ | <center>[[#toc|Back to Top]]</center> | ||
+ | |||
+ | === Geometric series === | ||
+ | <center><asy>defaultpen(linewidth(0.7)); unitsize(15); | ||
+ | int n = 10; /* # of iterations */ | ||
+ | real s = 6; /* square size */ | ||
+ | pair shiftR = (s+2,0); pen sm = fontsize(10); | ||
+ | void fillrect(pair A, pair B = (0,0), pen p = invisible, pen l = linewidth(1)){ filldraw(A--(A.x,B.y)--B--(B.x,A.y)--cycle, p, l); } | ||
+ | void htick(pair A, pair B, pair ticklength = (0,0.15)){ draw(A--B ^^ A-ticklength--A+ticklength ^^ B-ticklength--B+ticklength); } | ||
+ | |||
+ | for(int i = 0; i < 2; ++i) /* left */ | ||
+ | fillrect((s/2^(ceil(i/2)),s/2^(floor(i/2)))); | ||
+ | for(int i = 0; i < n; ++i) /* right */ | ||
+ | fillrect(shiftR,shiftR + (s/2^(ceil(i/2)),s/2^(floor(i/2)))); | ||
+ | label("$\frac 12$",(s*3/4,s/2),sm); label("$\cdots$",(s*1/4,s/2),sm); | ||
+ | label("$\frac 12$",shiftR+(s*3/4,s/2),sm); label("$\cdots$",shiftR+(s*1/4,s/2),sm); | ||
+ | label("$\frac 14$",shiftR+(s*1/4,s*3/4),sm); label("$\frac 18$",shiftR+(s*3/8,s/4),sm); | ||
+ | htick((0,-1), (s,-1)); htick(shiftR + (0,-1), shiftR + (s,-1)); | ||
+ | label("$1$",(s/2,-1),S,sm); label("$1$",shiftR+(s/2,-1),S,sm); | ||
+ | </asy> | ||
+ | The infinite [[geometric series]] <math>\frac 12 + \frac {1}{2^2} + \frac {1}{2^3} + \cdots = 1</math>.<br><br> | ||
+ | </center> | ||
+ | |||
+ | <center><asy> defaultpen(linewidth(0.7)); unitsize(15); | ||
+ | int n = 4; real h = 2; pen colors[] = {rgb(0.8,0,0),rgb(0,0.8,0)}; | ||
+ | void drawTriGrid(real s){ | ||
+ | for(int i = 0; i < 4; ++i){ | ||
+ | draw( (-s*3/2,s*(3/2 - i)) -- (s*3/2,s*(3/2 - i)), linetype("2 2")); | ||
+ | draw( (s*(3/2 - i),-s*3/2) -- (s*(3/2 - i),s*3/2), linetype("2 2")); | ||
+ | } | ||
+ | } | ||
+ | void fillrect(pair A, pair B, pen p){ filldraw(A--(A.x,B.y)--B--(B.x,A.y)--cycle, p, linewidth(1)); } | ||
+ | |||
+ | for(int i = 0; i < n; ++i) { | ||
+ | fillrect( ((-1)^i*-h/3^i*(3/2),-h/3^i*(3/2)) , ((-1)^i*-h/3^i*(1/2),h/3^i*(3/2)) , colors[0]); | ||
+ | fillrect(-((-1)^i*-h/3^i*(3/2),-h/3^i*(3/2)) ,-((-1)^i*-h/3^i*(1/2),h/3^i*(3/2)) , colors[1]); | ||
+ | fillrect( (-h/3^i*(1/2),(-1)^i*h/3^i*(1/2)) , (h/3^i*(1/2),(-1)^i*h/3^i*(3/2)), colors[0]); | ||
+ | fillrect(-(-h/3^i*(1/2),(-1)^i*h/3^i*(1/2)) ,-(h/3^i*(1/2),(-1)^i*h/3^i*(3/2)), colors[1]); | ||
+ | drawTriGrid(h/3^i); | ||
+ | } | ||
+ | </asy><br> | ||
+ | |||
+ | The infinite [[geometric series]] <math>\frac 13 + \frac {1}{3^2} + \frac {1}{3^3} + \cdots = \frac 12</math>.<br><br> | ||
+ | </center> | ||
+ | |||
+ | <center><asy> defaultpen(linewidth(0.7)); unitsize(15); | ||
+ | int n = 10; real h = 6; pen colors[] = {rgb(0.9,0,0),rgb(0,0.9,0),rgb(0,0,0.9)}; | ||
+ | pair shiftR = (h+3,0); | ||
+ | |||
+ | void drawEquilaterals(pair A, real s){ | ||
+ | filldraw(A--A+s*expi(2*pi/3)--A+(-s,0)--cycle,colors[0]); | ||
+ | filldraw(A--A+s*expi(2*pi/3)--A+s*expi(1*pi/3)--cycle,colors[1]); | ||
+ | filldraw(A--A+s*expi(1*pi/3)--A+(s,0)--cycle,colors[2]); | ||
+ | } | ||
+ | |||
+ | for(int i = 0; i < n; ++i) | ||
+ | drawEquilaterals(shiftR + (0,h-h/(2^i) ), (h/(2^(i+1))) *2/3^.5); | ||
+ | drawEquilaterals((0,0), h/3^.5); draw((-h/3^.5,0)--(h/3^.5,0)--(0,h)--cycle); label("$\vdots$",(0,3/4*h)); | ||
+ | </asy><br> | ||
+ | |||
+ | The infinite [[geometric series]] <math>\frac 14 + \frac {1}{4^2} + \frac {1}{4^3} + \cdots = \frac 13</math>.<br><br> | ||
+ | </center> | ||
+ | |||
+ | <center><asy> defaultpen(linewidth(1)); unitsize(15); | ||
+ | int n = 8; /* number of layers */ | ||
+ | real h = 3; /* square height */ | ||
+ | pen colors[] = {rgb(0.8,0,0),rgb(0,0.8,0),rgb(0,0,0.8)}; | ||
+ | pair shiftL = (-3*h,0); /* amount to shift second square left by */ | ||
+ | |||
+ | void drawSquares(real s, pair A = (0,0)){ | ||
+ | filldraw(shift(A)*shift(-2*s, -s)*xscale(s)*yscale(s)*unitsquare,colors[0]); | ||
+ | filldraw(shift(A)*shift(-2*s,-2*s)*xscale(s)*yscale(s)*unitsquare,colors[1]); | ||
+ | filldraw(shift(A)*shift(-s ,-2*s)*xscale(s)*yscale(s)*unitsquare,colors[2]); | ||
+ | } | ||
+ | for(int i = 0; i < n; ++i) | ||
+ | drawSquares(h/2^i); | ||
+ | drawSquares(h,shiftL); draw(shift(shiftL+(-2*h,-2*h))*xscale(2*h)*yscale(2*h)*unitsquare); | ||
+ | label("$\cdots$",shiftL+(-h/2,-h/2)); | ||
+ | </asy><br> | ||
+ | |||
+ | Another proof of the identity <math>\frac 14 + \frac {1}{4^2} + \frac {1}{4^3} + \cdots = \frac 13</math>. <br><br> | ||
+ | </center> | ||
+ | |||
+ | <center><asy> unitsize(15); defaultpen(linewidth(1)); pen sm = fontsize(10); | ||
+ | real r = 0.7, h = 4.5, n = 10, xsum = 0; | ||
+ | void htick(pair A, pair B, pair ticklength = (0,0.15)){ draw(A--B ^^ A-ticklength--A+ticklength ^^ B-ticklength--B+ticklength); } | ||
+ | |||
+ | filldraw(xscale(h)*yscale(h)*unitsquare,rgb(0.9,1,0.9)); draw((0,0)--(h/(1-r),0)--(0,h)); | ||
+ | for(int i = 0; i < n; ++i){ | ||
+ | xsum += r^i; | ||
+ | draw((h*xsum,0)--(h*xsum,h*(1-(1-r)*xsum))); | ||
+ | htick((h*(xsum-r^i),-1),(h*xsum,-1)); | ||
+ | if(i < 6) | ||
+ | label("$r^"+(string) i+"$",(h*(xsum-r^i/2),-1),S,sm); | ||
+ | else if(i == 8) | ||
+ | label("$\cdots$",(h*(xsum-r^i/2),-1.2),S,sm); | ||
+ | } | ||
+ | |||
+ | /* htick((-1,0),(-1,h),(.15,0)); htick((0,h+1),(h,h+1)); */ htick((h+1,h),(h+1,h*r),(.15,0)); | ||
+ | label("$1$",(0,h/2),W,sm); label("$1$",(h/2,h),N,sm); label("$1-r$",(h+1,h*(1+r)/2),E,sm); | ||
+ | </asy><br><br> | ||
+ | |||
+ | The infinite [[geometric series]] <math>\sum_{n=0}^{\infty} r^n = \frac{1}{1-r}</math>.<br><br></center> | ||
+ | |||
+ | <center><asy> unitsize(15); defaultpen(linewidth(1)); pen sm = fontsize(10); | ||
+ | real r = 0.55, h = 2.5, n = 7, xsum = 0; pair shiftD = -(0,h*r/(1-r)+2.5); | ||
+ | void htick(pair A, pair B, pair ticklength = (0,0.15)){ draw(A--B ^^ A-ticklength--A+ticklength ^^ B-ticklength--B+ticklength); } | ||
+ | |||
+ | draw((0,h*r/(1-r))--(0,0)--(h*n,0)); | ||
+ | for(int i = 1; i < n+1; ++i){ | ||
+ | draw((h*i,h*(r/(1-r)-xsum-r^(i)))--(h*i,h*(r/(1-r)-xsum))--(0,h*(r/(1-r)-xsum))); | ||
+ | if(i < 4) | ||
+ | label("$r^"+(string) i+"$", (0,h*(r/(1-r)-xsum-r^(i)/2)), W, sm); | ||
+ | htick((h*i,-1),(h*(i-1),-1)); | ||
+ | if(i < n) | ||
+ | label("$1$",(h*(i-1/2),-1),S,sm); | ||
+ | else if(i == n) | ||
+ | label("$\cdots$",(h*(i-1/2),-1.2),S,sm); | ||
+ | xsum += r^i; | ||
+ | } | ||
+ | draw((0,h*r/(1-r))+shiftD--shiftD--(h*n,0)+shiftD); | ||
+ | xsum = 0; | ||
+ | for(int i = 1; i < n+1; ++i){ | ||
+ | draw(shiftD+(h*i,0)--shiftD+(h*i,h*(r/(1-r)-xsum))--shiftD+(h*(i-1),h*(r/(1-r)-xsum))); | ||
+ | draw(shiftD+(h*i,h*(r/(1-r)-xsum))--shiftD+(0,h*(r/(1-r)-xsum)),linetype("4 4")+linewidth(0.5)); | ||
+ | if(i < 4) | ||
+ | label("$r^"+(string) i+"$", shiftD+(h*i,h*(r/(1-r)-xsum-r^(i)/2)), ENE, sm); | ||
+ | htick(shiftD+(h*i,-1),shiftD+(h*(i-1),-1)); | ||
+ | if(i < n) | ||
+ | label("$1$",shiftD+(h*(i-1/2),-1),S,sm); | ||
+ | else if(i == n) | ||
+ | label("$\cdots$",shiftD+(h*(i-1/2),-1.2),S,sm); | ||
+ | xsum += r^i; | ||
+ | } | ||
+ | </asy><br><br> | ||
+ | The [[arithmetic-geometric series]] <math>\sum_{n=1}^{\infty} nr^n = \sum_{n=1}^{\infty} \sum_{i=n}^{\infty} r^i = \sum_{n=1}^{\infty} \frac{r^{-n}}{1-r} = \frac{r}{(1-r)^2}</math>, also known as Gabriel's staircase.{{ref|2}}<br><br></center> | ||
+ | |||
+ | <center>[[#toc|Back to Top]]</center> | ||
== Geometry == | == Geometry == |
Revision as of 22:53, 22 January 2011
The following demonstrate proofs of various identities and theorems using pictures, inspired from this gallery.
Summations
The sum of the first odd natural numbers is .
The sum of the first positive integers is .
The sum of the first positive integers is .[1]
Nichomauss' Theorem: can be written as the sum of consecutive integers, and consequently that .
Another proof of the identity .
The identity , where is the th Fibonacci number.
Geometric series
The infinite geometric series .
The infinite geometric series .
The infinite geometric series .
Another proof of the identity .
The infinite geometric series .
The arithmetic-geometric series , also known as Gabriel's staircase.[2]
Geometry
The Pythagorean Theorem (first of many proofs): the left diagram shows that , and the right diagram shows a second proof by re-arranging the first diagram (the area of the shaded part is equal to , but it is also the re-arranged version of the oblique square, which has area ).[3]
Another proof of the Pythagorean Theorem (animated version).
Another proof of the Pythagorean Theorem; the left-hand diagram suggests the identity , and the right-hand diagram offers another re-arrangement proof.
COMING: The last (sixth) proof of the Pythagorean Theorem we shall present on this page, this one by dissection.
The smallest distance necessary to travel between , the x-axis, and then for is given by .[4]
In trapezoid with , then .
Miscellaneous
The Root-Mean Square-Arithmetic Mean-Geometric Mean inequality, .
The Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality.[5]
Fermat's Little Theorem: for (above ).
References
- ^ MathOverflow
- ^ Wolfram MathWorld
- ^ Attributed to the Chinese text Zhou Bi Suan Jing.
- ^ This is more of a proof without words of the AM-GM inequality ; though the lengths of the segments labeled RMS and HM can easily be verified to have values of , respectively, it might not be obvious from the diagram. It still serves as a useful graphical demonstration of the inequality.