The third is {(-1)^{n+1} }We write a general theoretical sequence as t_{1},t_{2},t_{3},... or {t_{n}}.
n > N => |t_{n} - 1|< eWe say that the limit of t_{n} = 1 and we write lim t_{n} = 1.
lim t_{n} = b <=> With each strictly positive real number e, corresponds a suitable positive integer N such that n > N => |t_{n} - b|< e .We say that the sequence converges to b.
The sequence {t_{n}} converges <=> With each strictly positive real number e, corresponds a suitable positive integer N such that n > N => |t_{n+p} - t_{n}|< e for each p=1,2,3,...Proof:
First part: suppose the sequence {t_{n}} converges to a number b.
A base environment of b, contains all the terms of {t_{n}},
starting from a suitable term.
From this it follows that
With each strictly positive real number e, corresponds a suitable positive integer N such that n > N => |t_{n+p} - t_{n}|< e for each p=1,2,3,...Second part: suppose that
With each strictly positive real number e, corresponds a suitable positive integer N such that n > N => |t_{n+p} - t_{n}|< e for each p=1,2,3,...Choose a fixed value of n. We divide all real numbers in two sets.
From |t_{n+p} - t_{n}|< e, we see that the number t_{n} -e is exceeded by an infinitely number of elements of the sequence and belongs to set A.
From |t_{n+p} - t_{n}|< e, we see that the number t_{n} +e is exceeded by a finite number of elements of the sequence and belongs to set B.
Therefore |t_{n} - b| =< e.
Now, t_{n+p} - b = t_{n+p} - t_{n} + t_{n} - b => |t_{n+p} - b| =< |t_{n+p} - t_{n}| + |t_{n} - b| < 2e for p=1,2,3,... => | t_{m} - b | < 2e for all m > n => The sequence {t_{n}} converges
n > N => t_{n} > rWe say that t_{n} has an infinite limit and we write lim t_{n} = infinity.
lim t_{n} = +infty <=> With each real number r, corresponds a suitable positive integer N such that n > N => t_{n} > r . lim t_{n} = -infty <=> With each real number r, corresponds a suitable positive integer N such that n > N => t_{n} < r .We say that the sequence diverges to infinity.
A real number M is an upper bound for {t_{n}} if and only if for each n : M >= t_{n} A real number m is an lower bound for {t_{n}} if and only if for each n : m =< t_{n} A sequence {t_{n}} is bounded if and only if {t_{n}} has an upper and a lower bound
If lim t_{n} = b and lim t_{n}' = b and if for all n > N : t_{n} =< t"_n =< t_{n}' Then lim t"_n = b
If lim t_{n} = b and if for all n > N : t_{n} = t_{n}' Then lim t_{n}' = b
If lim t_{n} = +infty and if for all n > N : t_{n} =< t_{n}' Then lim t_{n}' = +infty
If lim t_{n} = -infty and if for all n > N : t_{n} >= t_{n}' Then lim t_{n}' = -infty
If lim t_{n} = b > 0 Then t_{n} > 0 for all n starting from a suitable n=N
If lim t_{n} = b < 0 Then t_{n} < 0 for all n starting from a suitable n=N
If lim t_{n} = b Then lim |t_{n}| = |b|
If lim t_{n} = b is a real number Then, lim r.t_{n} = r.lim t_{n} lim 1/t_{n} = 1/lim t_{n} (if b not 0) lim t_{n}^{n} = ( lim t_{n} )^{n} lim t_{n}^{1/p} = ( lim t_{n} )^{1/n} (if both sides exist)
If lim t_{n} and lim t_{n}' are real numbers, Then, lim (t_{n} + t_{n}') = lim t_{n} + lim t_{n}' lim (t_{n} - t_{n}') = lim t_{n} - lim t_{n}' lim t_{n}.t_{n}' = lim t_{n} . lim t_{n}' t_{n} lim t_{n} lim ----- = ------------ (if both sides exist) t_{n}' lim t_{n}'
-(+infty)=-infty -(-infty)=+infty +(-infty)=-infty +(+infty)=+infty |+infty|= +infty |-infty|= +infty (+infty)^{n} = +infty (-infty)^{2n}= +infty (-infty)^{2n+1}= -infty nth-root(+infty)=+infty (2n+1)th-root(-infty)=-infty for each r = strictly positive real number r(+infty)=+infty r(-infty)=-infty -r(+infty)=-infty -r(-infty)=+infty for each real number r r/+infty = 0 r/-infty = 0 +infty + r = +infty -infty + r = -infty r - infty = -infty r + infty = +infty +infty +(+infty)= +infty -infty +(-infty)= -infty +infty -(-infty)= +infty -infty -(+infty)= -infty (+infty)(+infty)= +infty (-infty)(+infty)= -infty (+infty)(-infty)= -infty (-infty)(-infty)= +infty
lim t_{n} = +infty => lim (-t_{n}) = -infty lim t_{n} = -infty => lim (-t_{n}) = +infty lim t_{n} = +infty => lim |t_{n}| = +infty lim t_{n} = -infty => lim |t_{n}| = +infty if lim t_{n} = +infty or lim t_{n} = -infty, then lim t_{n}^{p} = (lim t_{n})^{p} lim (nth-root(t_{n})) = nth-root(lim t_{n}) (if both sides exist) lim (c.t_{n}) = c.(lim t_{n}) (with c real number) lim (c/t_{n}) = 0 (with c real number) lim t_{n} = +0 => lim (1/t_{n})=+infty lim t_{n} = -0 => lim (1/t_{n})=-infty if lim t_{n} = infty and lim t_{n}'= b (real and not zero) then lim(t_{n}+t_{n}')=lim t_{n} +lim t_{n}' lim(t_{n}-t_{n}')=lim t_{n} -lim t_{n}' lim(t_{n}'-t_{n})=lim t_{n}' -lim t_{n} lim(t_{n}'.t_{n})=lim t_{n}' .lim t_{n} lim(t_{n}/t_{n}')=lim t_{n} / lim t_{n}' lim(t_{n}'/t_{n})= 0 lim t_{n} = +infty and lim t_{n}' = +infty => lim(t_{n}+t_{n}')=lim t_{n} + lim t_{n}' lim t_{n} = -infty and lim t_{n}' = -infty => lim(t_{n}+t_{n}')=lim t_{n} + lim t_{n}' lim t_{n} = +infty and lim t_{n}' = -infty => lim(t_{n}-t_{n}')=lim t_{n} - lim t_{n}' lim t_{n} = -infty and lim t_{n}' = +infty => lim(t_{n}-t_{n}')=lim t_{n} - lim t_{n}' lim t_{n} = infty and lim t_{n}' = infty => lim(t_{n}.t_{n}')=lim t_{n} . lim t_{n}'
t_{n} = t_{1} + (n-1).vWith each choice of t_{1} corresponds exactly one sequence .
S = t_{1} + t_{1} + v + t_{1} + 2.v + ... + t_{n} Now write the same sequence in reverse order S = t_{n} + t_{n} - v + t_{n} - 2.v + ... + t_{1} Addition gives 2.S = (t_{1} + t_{n}).n So, (t_{1} + t_{n}).n S = ---------------- 2
The sum of the four roots is zero. So, we can write the roots as ( -3h, -h, h, 3h ). Then we have
(x+3h)(x-h)(x+h)(x-3h) = x^{4} - (1+3m) x^{2} + m^{2} <=> (x^{4} - 9h^{2})(x^{2}-h^{2}) = x^{4} - (1+3m) x^{2} + m^{2} <=> 9h^{4} = m^{2} 10h^{2} = 1 + 3m Case 1 : m = 3h^{2} h^{2} = 1 m = 3 Case 2 : m = -3h^{2} 19h^{2} = 1 m = -3/19
t_{n} = t(n-1).qWith each choice of t_{1} corresponds exactly one sequence .
for all n > 1 : If q = 1 + x > 1 , then q^{n} > 1 + n.x (1)Proof:
The theorem holds for n = 2. (2)
q^{k} > 1 + k.x => q.q^{k} > (1+x).(1 + k.x) => q^{k+1} > 1 + (k + 1)x + k.x.x => q^{k+1} > 1 + (k + 1)x (3)From (2) and (3) it follows that (1) holds for all n >1.
If q > 1 then q^{n} is rising and has no upper bound. lim q^{n} = +infty
1 1 n (---) > 1 and lim (---) = +infty |q| |q| Hence, n 1 1 lim |q | = lim -------- = --------- = 0 1 n +infty (---) |q|
t_{n} = t_{1}.q^{n-1} = t_{1}.q^{n} /q = (constant number) .q^{n} If q > 1 then lim t_{n} = + infty or -infty If q = 1 then the sequence is constant lim t_{n} = t_{1} If 0 < q <1 then lim t_{n} = 0 If -1< q <0 then lim t_{n} = 0 If q = -1 then lim t_{n} don't exist If q < -1 then lim t_{n} don't exist
S = t_{1} + t_{2} + ... + t_{n} , then => S.q = t_{1}.q + t_{2}.q + ... + t_{n}.q => S.q = t_{2} + ... + t_{n} + t_{n}.q => S.q - S = t_{n}.q - t_{1} => S(q-1) = t_{n}.q - t_{1} t_{n}.q - t_{1} t_{1}.q^{n} - t_{1} => S = ---------------- = ---------------- (q - 1) (q - 1) t_{1}.(q^{n} - 1) => S = ---------------- (q - 1)
Take 0 < |q| < 1 t_{1}.q^{n} - t_{1} t_{1} S = lim ---------------- = ----------- (q - 1) (1 - q)
We give an example. Say you want to calculate the sum
S = 1.2^{-1} + 2.2^{-2} + 3.2^{-3} + ... + n.2^{-n}
We write S as follows:
2^{-1} 2^{-2} + 2^{-2} 2^{-3} + 2^{-3} + 2^{-3} 2^{-4} + 2^{-4} + 2^{-4} + 2^{-4} . . . . . 2^{-n} + 2^{-n} + 2^{-n} + 2^{-n} + ... +2^{-n}Each column is the sum of terms of a geometric sequence with ratio 1/2.
The sum of the first column is, after simplification, 1 - 2^{-n} The sum of the second column is, after simplification, 2^{-1} - 2^{-n} The sum of the third column is, after simplification, 2^{-2} - 2^{-n} The sum of the fourth column is, after simplification, 2^{-3} - 2^{-n} ...... The sum of the n-th column is, after simplification, 2^{1-n} - 2^{-n}S = ( 1 + 2^{-1} + 2^{-2} + 2^{-3} + ... + 2^{1-n} ) - n.2^{-n}
The first part is again a sum of n terms of a geometric sequence. We find after simplification:
S = 2 - 2^{1-n} - n 2^{-n}
This was the ultimate goal. The sum of n terms of a non-geometric row was calculated using the properties of a geometric sequence.