successions

Si a_{n+m} = a_{n}+a_{m} ==> (a_{n}/n) = a_{1}

a_{n} = a_{1+...(n)...+1} = n·a_{1}

Si ( a_{n+m} = a_{n}+a_{m} & a_{1} >] 0 ) ==> a_{n} és creishent

Si ( a_{n+m} = a_{n}+a_{m} & a_{1} [< 0 ) ==> a_{n} és decreishent

a_{n+1} = a_{n}+a_{1} >] a_{n}

a_{n+1} = a_{n}+a_{1} [< a_{n}

successions monotonia

Si ( a_{n} >] 0 & e^{a_{n}} [< a_{n+1} ) ==> a_{n} és creishent


a_{n} [< 1+a_{n} [< e^{a_{n}} [< a_{n+1}


Si ( a_{n} >] 0 & (-1)·e^{a_{n}} >] a_{n+1} ) ==> a_{n} és decreishent


a_{n} >] (-1)+a_{n} >] (-1)·e^{a_{n}} >] a_{n+1}

series numériques criteri

teorema:
Si [∀k][ k€N ==> [∃s][ s > 0 & a_{k} [< s^{k} < 1 & (a_{k}/a_{k+1}) [< a_{k+(-1)} ] ] ==> ...
... ∑ a_{k} [< ( 1/( 1+(-1)·s^{2} ) ) [< oo


demostració:
(a_{k}/a_{k+1}) [< a_{k+(-1)}
a_{k} [< a_{k+1}a_{k+(-1)} [< s^{k+1}s^{k+(-1)} = s^{2k}


∑ a_{k} [< ∑ s^{2k} [< ( 1/( 1+(-1)·s^{2} ) ) [< oo


teorema:
Si [∀k][ k€N ==> [∃s][ s > 0 & a_{k} [< s^{k} < 1 & (a_{k}/a_{k+p}) [< a_{k+(-p)} ] ] ==> ...
... ∑ a_{k} [< ( 1/( 1+(-1)·s^{2} ) ) [< oo

series numériques criteri del quocient

teorema:
Si [∀k][ k€N ==> [∃s][ s > 0 & a_{1} [< 1 & ( a_{k+1}/a_{k} ) [< s [< ( 1+(-1)(1/oo) ) ] ] ==> ...
... ∑ a_{k} [< oo


demostració:
( a_{k+1}/a_{k} ) [< s
a_{k+1} [< s·a_{k}
a_{k+1} [< s^{k+1}
∑ a_{k} [< ∑ s^{k} [< ( 1/(1+(-s)) ) [< oo


teorema:
( k/(k+1) ) = ( 1+( (-1)/(k+1) ) ) [< ( 1+(-1)(1/oo) ) = s


∑ ( 1/k ) [< oo

series numériques criteri de la arrel enéssima

teorema:
Si [∀k][ k€N ==> [∃s][ s > 0 & ( a_{k} )^{(1/k)} [< s [< ( 1+(-1)(1/oo) ) ] ] ==> ∑ a_{k} [< oo


demostració:
∑ a_{k} [< ∑ s^{k} [< ( 1/(1+(-s)) ) [< oo

limits de successions

lim ( oo+(-n) ) = 0
( oo+(-oo) ) = 0


lim ( (oo+p)+(-n) ) = p
( (oo+p)+(-oo) ) = p


lim ( (oo^{q}+p)+(-1)·n^{q} ) = p
( (oo^{q}+p)+(-1)·oo^{q}) ) = p


lim ( ( (oo+1)^{2}+(-1)(n^{2}+1) )/n ) = 2
( ( (oo+1)^{2}+(-1)(oo^{2}+1) )/oo ) = 2


lim ( ( (oo+1)^{2}+(-1)(n+(-1))^{2} )/n ) = 4
( ( (oo+1)^{2}+(-1)(oo+(-1))^{2} )/oo ) = 4


lim ( ( (oo+1)^{2}+(-1)(n+1) )/(n^{2}+n) ) = 1
( ( (oo+1)^{2}+(-1)(oo+1) )/(oo^{2}+oo) ) = 1


lim ( ( (oo+1)^{3}+(-1)(n^{3}+1) )/(n^{2}+n) ) = 3
( ( (oo+1)^{3}+(-1)(oo^{3}+1) )/(oo^{2}+oo) ) = 3


lim ( ( (oo+1)^{4}+(-1)(n^{2}+1)^{2} )/(n^{3}+n^{2}+n) ) = 4
( ( (oo+1)^{4}+(-1)(oo^{2}+1)^{2} )/(oo^{3}+oo^{2}+oo) ) = 4

progressions aritmética, geométrica y potencial

( a_{1}=b & a_{n+1} = a_{n}+b ) <==> a_{n+1}=(n+1)·b


( a_{1}=b & a_{n+1} = b·a_{n} ) <==> a_{n+1}=b^{(n+1)}


( a_{1}=b & a_{n+1} = (a_{n})^{m} ) <==> a_{n+1}=b^{m^{n+1}}

successions

si ( m >] 0 & a_{n}+m [< a_{n+1} ) ==> a_{n} és creishent
a_{n} [< a_{n}+m [< a_{n+1}


si ( m [< 0 & a_{n}+m >] a_{n+1} ) ==> a_{n} és decreishent
a_{n} >] a_{n}+m >] a_{n+1}

succesions

Si ( a_{1} >] 0 & a_{n+m} >] a_{n}+a_{m} ) ==> a_{n} és creishent
a_{n+1} >] a_{n}+a_{1} >] a_{n}


Si ( a_{1} [< 0 & a_{n+m} [< a_{n}+a_{m} ) ==> a_{n} és decreishent
a_{n+1} [< a_{n}+a_{1} [< a_{n}


Si a_{n+m} >] a_{n}+a_{m} ==> (a_{n}/n) >] a_{1}
a_{n} >] a_{1}+...(n)...+a_{1} = na_{1}


Si a_{n+m} [< a_{n}+a_{m} ==> (a_{n}/n) [< a_{1}
a_{n} [< a_{1}+...(n)...+a_{1} = na_{1}


Si ( n >] a_{n} & a_{n+m} >] a_{n}+a_{m} ) ==> 1 >] (a_{n}/n) >] a_{1}
n >] a_{n}


Si ( n [< a_{n} & a_{n+m} [< a_{n}+a_{m} ) ==> 1 [< (a_{n}/n) [< a_{1}
n [< a_{n}

series númeriques

teorema:
Si ( a_{n} >] 0 & ∑ ( n·a_{n} ) < oo ) ==> ∑ ( a_{n} ) < oo


demostració:
∑ ( a_{n} ) [< ∑ ( n·a_{n} ) < oo


teorema:
Si ( a_{n} >] 0 & ∑ ( a_{n} ) < oo ) ==> ∑ ( (1/n)·a_{n} ) < oo


demostració:
∑ ( (1/n)·a_{n} ) [< ∑ ( a_{n} ) < oo

series numériques

teorema:
Si 0 < a_{n} < 1 ==> ∑ ( a_{n} ) < oo


demostració:
∑ ( a_{n} ) < 1+...(oo)...+1 = oo


teorema:
Si ( m€N & 0 < a_{n} < m ) ==> (1/m)·∑ ( a_{n} ) < oo


demostració:
∑ ( a_{n} ) < m+...(oo)...+m = m·oo

series numériques

teorema:
Si ( a_{n} >] 0 &  m€N & ∑ ( m·a_{n} ) < oo ) ==> ∑ ( a_{n} ) < oo


demostració:
m·∑ ( a_{n} ) = ∑ ( m·a_{n} ) < oo


∑ ( a_{n} ) < (oo/m) < oo


teorema:
Si ( a_{n} >] 0 & ∑ ( oo·a_{n} ) < oo ) ==> ∑ ( a_{n} ) < 1


demostració:
oo·∑ ( a_{n} ) = ∑ ( oo·a_{n} ) < oo


∑ ( a_{n} ) < (oo/oo) = 1

series númeriques mitjes

teorema:
Si ( a_{n} >] 0 & b_{n} >] 0 &  ∑ ( a_{n} ) < oo & ∑ ( b_{n} ) < oo ) ==> ...
... ∑ ( (a_{n}+b_{n})/2 ) < oo


demostració:
∑ ( a_{n}+b_{n} ) = ∑ ( a_{n} ) + ∑ ( b_{n} )  < 2·oo


teorema:
Si ( a_{n} >] 0 & b_{n} >] 0 & ∑ ( a_{n} ) < oo & ∑ ( b_{n} ) < oo ) ==> ...
... ∑ ( (a_{n}·b_{n})^{(1/2)} ) < oo


demostració:
∑ ( (a_{n}·b_{n})^{(1/2)} ) [< ∑ ( (a_{n}+b_{n})/2 ) < oo

índex de matemàtiques

index de etiquetes matemàtiques